PX282 - J9 - moons

mercury and venus: no moons
earth: a single large moon, believed to have been formed from a giant impact
mars: two small captured asteroids - phobos and deimos
phobos orbits faster than mars spins (0.3 days vs ~1 day)
the tidal synchronization is spinning up mars and causing the orbit of phobos to decay
phobos will eventually break apart when it fills its roche lobe/hill sphere

roche limit

image: Carrol & Ostile, An Introduction to Modern Astrophysics, 2007

it is the limit to orbital separation, where the self gravity is smaller than the differential gravity from the planet

d F_{G} = - \frac{2 G M m}{r^{3}} d r

\frac{G M_{m} m}{R_{m}^{2}} \leq \frac{2 G M_{p} m}{a^{3}} R_{m}

where,
$M_{m}$ is the mass of the moon,
$M_{p}$ is the mass of the planet,
$m$ is the mass of a test particle,
$R_{m}$ is the radius of the moon,
and $a$ is the orbital separation

the roche limit:

a_{R}^{3} = 2 \frac{M_{p}}{M_{m}} R_{m}^{3} ⟹ a_{R} = 2^{\frac{1}{3}} {(\frac{ρ_{p}}{ρ_{m}})}^{1 / 3} R_{p}

where, $M = 4 / 3 π R^{3} ρ$

this is for a rigid body that maintains its shape
for a fluid body (in hydrostatic equilibrium), the the pre-factor of $2$ changes to $2.456$
slow rotation of mercury and venus means that any moon within their hill spheres would spiral in, and eventually be destroyed at its roche limit
also, their proximity to the sun means very small hill spheres