  An intriguing question in the context of dynamics arises: Could a moon
possess a moon itself? Such a configuration does not exist in the Solar System,
although this may be possible in theory. Kollmeier et al. (2019) determined the
critical size of a satellite necessary to host a long-lived sub-satellite, or
submoon. However, the orbital constraints for these submoons to exist are still
undetermined. Domingos et al. (2006) indicated that moons are stable out to a
fraction of the host planet Hill radius $R_{H,p}$, which in turn depends on the
eccentricity of its host's orbit. Motivated by this, we simulate a system of
exomoons and submoons for $10^5$ planetary orbits, while considering many
initial orbital phases to obtain the critical semimajor axis in terms of
$R_{H,p}$ or the hosts satellite's Hill radius $R_{H,sat}$, respectively. We
find that, assuming circular coplanar orbits, the stability limit for exomoons
is 0.40 $R_{H,p}$ and for a submoon is 0.33 $R_{H,sat}$. Additionally, we
discuss the observational feasibility of detecting these sub-satellites through
photometric, radial velocity, or direct imaging observations using the
Neptunes-sized exomoon candidates Kepler 1625b-I (Teachey et al. 2018) and
identify how stability can shape the identification of future candidates.
