Exomoons: Natural Satellites of Exoplanets

• Exomoons are natural satellites orbiting exoplanets or planetary-mass bodies outside the Solar System, detectable through gravitational effects, photometric, or spectroastrometric methods and characterized by stellar, planetary, and satellite masses and orbits.
• Astronomers use transit timing variations (TTV), transit duration variations (TDV), and transit radius variations (TRV), alongside photometric or direct detection methods to identify exomoons, although none have been unambiguously confirmed yet.
• A key objective for astronauts is understanding exomoon formation, particularly in Kepler-1625 b-i and Kepler-1708 b-i systems, as habitable exomoons can enhance our understanding of satellite formation, migration, and the wider dynamics of planetary systems.

An exomoon is a natural satellite orbiting an exoplanet or another planetary-mass body outside the Solar System. Exomoons are inferred through their gravitational effects on host-planet transits, direct photometric or spectroastrometric signatures, radio modulation, phase-curve contributions, or Doppler motion of the host planet. No exomoon has been confirmed unambiguously; Kepler-1625 b-i and Kepler-1708 b-i remain candidates supported by different combinations of transit timing and photometric evidence (Teachey et al., 2018, Kipping et al., 2022). Their scientific importance extends from satellite formation and tidal evolution to planetary interiors, climate dynamics, and habitability.

1. Physical definition and dynamical architecture

An exomoon is a gravitationally bound satellite of an exoplanet. In the standard hierarchical approximation, the planet and moon form a local two-body system whose barycentre follows an orbit around the star. The planet and moon separately orbit their common barycentre, so the planet executes a reflex motion that can be observed even when the moon itself does not measurably transit the star (Kipping, 2011).

The principal parameters are the stellar, planetary, and satellite masses, $M_*$, $M_P$, and $M_S$; stellar and planetary radii, $R_*$ and $R_P$; the planet–star or planet–moon-barycentre semimajor axis, $a_P$; the satellite semimajor axis about the barycentre, $a_S$; the corresponding periods, $P_P$ and $P_S$; eccentricities, $e_P$ and $M_P$0; inclinations, $M_P$1 and $M_P$2; arguments of periapsis, $M_P$3 and $M_P$4; and the satellite longitude of ascending node, $M_P$5. A useful orientation combination is

$M_P$6

For a satellite within the Hill sphere, the nested two-body approximation gives

$M_P$7

where $M_P$8 is the satellite distance in Hill-radius units. Under $M_P$9, Kepler’s third law gives

$M_S$0

Thus, for a bound moon with $M_S$1, $M_S$2. Transit observations sample the satellite only once per planetary orbit, so the lunar signal is undersampled and generally aliased.

A satellite orbit must lie between an inner disruption boundary and an outer stellar-perturbation boundary. The Roche limit is approximately

$M_S$3

where $M_S$4 and $M_S$5 are planetary and lunar mean densities. The outer region is commonly represented as a fraction of the Hill radius,

$M_S$6

with stable prograde orbits often approximated by $M_S$7, where $M_S$8 is commonly taken near $M_S$9, or more conservatively near $R_*$0$$a_{M,\max}=0.4895R_H \left(1.0000-1.0305e_P-0.2738e_M\right)</p> <p>$$R_*$1</p> <p>a_{PB}=\frac{M_S}{M_P}a_S</p> <p>$R_*$2</p> <p>\delta_{\rm TTV}\propto M_Sa_S.</p> <p>$R_*$$3</p> <p>\delta_{\rm TDV-V}\propto M_Sa_S^{-1/2},</p> <p>$$R_*$$4</p> <p>\eta\equiv \frac{\delta_{\rm TDV-V}}{\delta_{\rm TTV}} \simeq \frac{\tilde T_B}{2\pi P_S}</p> <p>$$R_*$$5</p> <p>\Lambda_{\rm TTV}=\cos f_S,\qquad \Lambda_{\rm TDV-V}=\sin f_S,</p> <p>$$R_*$6</p> <p>\Delta\phi=\frac{\pi}{2}.</p> <p>$R_*$7</p> <p>{\rm TDV-TIP}\propto M_Sa_S,</p> <p>$R_*$$8</p> <p>\left(\frac{\rm TTV}{A_{\rm TTV}}\right)²⁺ \left(\frac{\rm TDV}{A_{\rm TDV}}\right)^2=1.</p> <p>$$R_*$$9</p> <p>P_{\rm TTV}= \frac{1}{ \left| \frac{1}{P_S}- \frac{{\rm round}(P_P/P_S)}{P_P} \right|}.</p> <p>$$R_P$$0</p> <p>\mathbf c(\lambda)=\frac{\mathbf r_P}{d}+f_M(\lambda)\boldsymbol\beta.</p> <p>$$R_P$$1</p> <p>K_P= \left[ \frac{2\pi G}{P_M(1-e_M^{2)^{3/2}}} \frac{M_M^{3\sin^{3i_M}{(M_P+M_M)^2}}} \right]^{1/3}.</p> <p>$$R_P$$2</p> <p>K_P\propto M_M\,M_P^{{-2/3}\,P_M^{-1/3}\sin} i_M.</p> <p>$$R_P$$3</p> <p>f_C=\frac{eB_{\rm pole}}{2\pi m_e}.</p> <p>$$R_P$$4</p> <p>R_S/R_P=0.263^{{+0.040}_{-0.042},}</p> <p>$$R_P$5</p> <p>F_{\rm RG}&gt;\bar F_S^{\rm glob}.</p> <p>$R_P$$6</p> <p>h_S\propto M_P^{2R_S^{3a_{PS}^{{-9}e_{PS}^2.}}}</p> <p>$$R_P$$7</p> <p>Q\sim3\times10^{5-3\times10⁶}</p> <p>$$R_P$$8</p> <p>Q\sim1.5\times10^{5-4\times10^5.}$$R_P$9, and the initial satellite orbit is generally unknown.

Formation scenarios include circumplanetary-disk accretion, giant impacts, capture, coalescence, Trojan capture, gas drag, and pull-down capture. Regular circumplanetary-disk systems are expected to have relatively low total satellite masses, whereas impacts and capture can generate a dominant massive moon. The large candidate moons associated with Kepler-1625 b and Kepler-1708 b would challenge ordinary in-situ satellite formation if confirmed.

High-eccentricity migration presents an additional constraint. In binary-induced von Zeipel–Lidov–Kozai migration, a sufficiently massive moon can induce apsidal precession that suppresses the planet’s eccentricity growth, an effect termed moon shielding. If the moon is too weak to prevent migration, it is generally stripped, driven into the planet, ejected, or collides with the star. Simulations found that only approximately 0.6% of low-mass and 9.1% of massive moons remained bound while efficiently shielding the planet, and essentially no systems produced a hot Jupiter that successfully retained its moon (Trani et al., 2020). A confirmed moon around a hot Jupiter would therefore challenge binary-driven high-eccentricity migration for that planet.

Theoretical studies also identify an observational tension: planets with short orbital periods are easiest to monitor for TTVs because they transit frequently, but they are poor long-term moon hosts. In a long-term tidal-survival survey, most planets with periods shorter than approximately 10 days had little or no stable satellite phase space. Survival increased between approximately 10 and 300 days and reached roughly 70%–90% for many systems beyond 300 days under the adopted distributions (Dobos et al., 2021). Long-period cool giant planets are consequently promising exomoon targets despite their sparse transit sampling.

Exomoons are thus relevant to several areas of planetary science. Their transit signals can constrain satellite mass, radius, orbit, and density; their tidal evolution can probe planetary $a_P$0, Love numbers, and interior structure; their climate can differ fundamentally from that of isolated planets because of synchronous rotation and planetary illumination; and their presence or absence can constrain formation and migration histories. Current candidates remain unconfirmed, and all numerical predictions depend on assumptions about noise, stellar activity, orbital stability, tidal dissipation, formation history, and atmospheric physics. Confirmation requires repeated, dynamically consistent observations using multiple diagnostics rather than a single transit anomaly or timing signal.