My main objective is to find the radius(or relative size) of the stars from the Gaia DR3 archive. However as I was unable to query the radius directly I am looking into the Stefan-Bolzmann Law which can derive radius from luminosity(L) and effective temperature(T): $$R = \sqrt{\frac{L}{4\pi\sigma T^4}}$$ I am finding effective temperature using the [Ballesteros equation][1] where I am setting B-V = bp_rp (difference between the blue photometric band and the red photometric band) from Gaia DR3 archive. But as for luminosity I can not find any equations that I can use where the data is available at Gaia DR3 archive. So, how can one find the relative size or luminosity of stars from the archive?
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1$\begingroup$ Related astronomy.stackexchange.com/questions/20702/… and astronomy.stackexchange.com/questions/48103/… $\endgroup$– ProfRobCommented Sep 18 at 14:56
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Radius and luminosity are no measured values.
The relation from the magnitudes measured in the individual bands to the overall luminosity is not straight forward as the colour-magnitude diagramme is a non-unique relation of B-V (and thus also B-R) against magnitude. It would be somewhat straight forward if you assume that your star is a main sequence star.
However, Gaia offers you the distance to the stars in terms of parallax. Thus with the measured magnitude and the distance you can calculate the absolute magnitudes, and from that you can get the luminosity.
E.g. to get the luminosity in solar units (with the absolute magnitude of the Sun of $m_{V,sun}=4.8$) and absolute magnitude $M_V$ of your object (I use here the V filter, but can be applied to any you like): $$L = 10^{0.4\cdot (m_{V,sun}-M_V)}$$
List of Gaia parameters and their explanation: https://gea.esac.esa.int/archive/documentation/GEDR3/index.html
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$\begingroup$ This isn't a formula that would give the luminosity. Gaia does not measure a V magnitude and even if it did, the relationship you propose only works for a star with the same spectrum as the Sun. $\endgroup$– ProfRobCommented Sep 18 at 14:53
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$\begingroup$ I used to think that apparent magnitude and distance gave luminosity, but I am guessing that extinction is a problem. $\endgroup$ Commented Sep 26 at 1:07
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$\begingroup$ Yes, extinction and the fact that a visual magnitude (which the Gaia catalogue does not give) will only directly give you the lumoinosity of a sun-like star. The error associated with the latter (e.g. applying the equation above to say a B-star or an M-type star) is considerable. @JackR.Woods $\endgroup$– ProfRobCommented Oct 18 at 9:40