Relationships between G magnitude, Johnson V magnitude, and spectral type of stars?

I'm simulating a population of binary stars for a summer research project. I'm adding uncertainties to my simulated observables such as angular position and radial (line of sight) velocity. I'm using astrometric and spectroscopic uncertainties from different telescopes to model what the data would look like if it were observed by different experiments. In my simulation, I'm including the fact that the values of the uncertainties for said observables depend on the magnitude (brightness) of the star. The fainter the star, the lower the precision with which we can measure its position, velocity, and so on. Makes sense.

Certain parameters of binary orbits such as the mass ratio distribution of the components depend on the spectral type of the primary, and I would like to include this in my simulation as well. In order to do this, I will generate a random spectral type for each orbit, and from there determine the mass of the primary, the mass ratio, and so on. As I understand it, spectral type, in general, is related to the magnitude of the star. My goal is, using the randomly generated spectral type (from a certain distribution) for each binary pair, to calculate the magnitude of the stars in order to assign the correct uncertainty in angular position and radial velocity.

One of the telescopes whose uncertainties I want to use to simulate observed data is GAIA. I came across this link: https://www.cosmos.esa.int/web/gaia/science-performance#astrometric%20performance and the associated paper: de Bruijne, J.H.J. Science performance of Gaia, ESA’s space-astrometry mission. Astrophys Space Sci 341, 31–41 (2012). https://doi.org/10.1007/s10509-012-1019-4, which include exactly what I'm looking for: the varying astrometric and spectroscopic uncertainties for stars of different magnitudes. The spectroscopic uncertainty depends on Johnson V magnitude, but the astrometric uncertainty varies with a different description of magnitude, a parameter $$G$$, e.g. $$G = 15$$ mag. I'm trying to relate these to spectral types, but when I researched it I wasn't able to find an explanation of $$G$$ mag anywhere.

I did read about absolute, apparent, and Johnson V magnitudes (https://en.wikipedia.org/wiki/Magnitude_(astronomy)), but I still don't understand where the parameter $$G$$ comes from, or the relationship between spectral type and magnitude as it relates to my simulation. Can anyone help me relate $$G$$, Johnson V magnitude, and spectral type? Or explain what is flawed about my simulation plan?

My apologies if this is better suited to the astronomy stack exchange, and thank you in advance.

The $$G$$ magnitude is the apparent magnitude of the star seen through the broadest of the Gaia photometric bandpasses.

The Gaia photometric band is much broader than a $$V$$ filter, so although $$G$$ and $$V$$ are monotonically related, the difference between the two does depend on the intrinsic colour/spectral type of the star and the extinction-related reddening of the observed spectrum.

Thus there is no single relationship that can be given.

There is nothing flawed in your plan, except that you don't seem to have thought about extinction and reddening (maybe these are all nearby stars?) and you just need some relationship between absolute $$G$$ magnitude and spectral type.

Such a relationship could be constructed for main sequence stars, although personally, I would not use spectral type as my base variable, I would use the primary mass.

The current properties of the Gaia EDR3 dataset are summarised at https://www.cosmos.esa.int/web/gaia/earlydr3

The definitions of the $$G$$ bandpass and magnitude zeropoint can be found here: https://www.cosmos.esa.int/web/gaia/edr3-passbands

Much more detail about the Gaia photometry can be found in Riello et al. (2020), including Fig.14 which shows $$G$$ errors vs $$G$$.

The EDR3 $$G$$ magnitudes are not that different to the magnitudes reported in DR2. So to get some sort of relationship between absolute $$G$$ magnitude and mass/spectral type, you will need to look at papers on plotting HR diagrams using the Gaia data (e.g. Babusiaux et al. 2018).

If you insist on using $$V$$ magnitudes, then an approximate (colour-dependent) conversion (valid for DR2 magnitudes) is given at https://gea.esac.esa.int/archive/documentation/GDR2/Data_processing/chap_cu5pho/sec_cu5pho_calibr/ssec_cu5pho_PhotTransf.html