  The recent Gaia Data Release 3 has unveiled a catalog of over eight hundred
thousand binary systems, providing orbital solutions for half of them. Since
most of them are unresolved astrometric binaries, several astrophysical
parameters that can be only derived from their relative orbits together with
spectroscopic data, such as the individual stellar masses, remain unknown.
Indeed, only the mass of the primary, $\texttt{m1}$, and a wide interval,
$\texttt{[m2_lower, m2_upper]}$, for the secondary companion of main-sequence
astrometric binaries have been derived to date (Gaia Collaboration et al.,
2023). In order to obtain the correct values for each component, we propose an
analytic algorithm to estimate the two most probable relative orbits and
magnitude differences of a certain main-sequence or subgiant astrometric binary
using all available Gaia data. Subsequently, both possible solutions are
constrained to the one that is consistent with $\texttt{m1, m2_lower}$ and
$\texttt{m2_upper}$. Moreover, we deduce not only the correct values of the
individual masses for each binary but also the size of the telescope necessary
to resolve their components. The workflow of our algorithm as well as the
ESMORGA (Ephemeris, Stellar Masses, and relative ORbits from GAia) catalog with
more than one hundred thousand individual masses, spectral types, and effective
temperatures derivated from its application are also presented.
