3.1. Evolution of planetary and satellite systems

There has been much discussion of the dynamical evolution of the planets and the Kuiper Belt (Batygin & Brown Reference Batygin and Brown2010; Tsiganis et al. Reference Tsiganis, Gomes, Morbidelli and Levison2005; Hahn & Malhotra Reference Hahn and Malhotra1999; Kokubo & Ida Reference Kokubo and Ida1995) including their formation by accretion of a circumsolar solids and gas disk (Pollack et al. Reference Pollack, Hubickyj, Bodenheimer, Lissauer, Podolak and Greenzweig1996; Alibert et al. Reference Alibert, Mordasini, Benz and Winisdoerffer2005; Bitsch et al. Reference Bitsch, Izidoro, Johansen, Raymond, Morbidelli, Lambrechts and Jacobson2019), the development of their orbital eccentricity, and their movement inwards or outwards due to an exchange of orbital angular momentum with planetesimals (Nesvorný Reference Nesvorný2018). Specifically, Soter (Reference Soter2006) concluded that ‘there is a tendency of disk evolution in a mature system to produce a small number of large bodies (planets) in non-intersecting or resonant orbits, which prevent collisions between them’.

Although none of these papers considers the formation and evolution of the systems of natural satellites on the planets, it might be assumed that related dynamics may be in play as those that formed the planets. The ‘gas-starved’ model of Canup & Ward (Reference Canup and Ward2002) and (Reference Canup and Ward2006) is the most successful model for the formation of the large satellites of the giant planets; this model has been summarised, and an alternative model proposed, by Batygin & Morbidelli (Reference Batygin and Morbidelli2020). It is likely that most of the inner, prograde planetary satellites have formed by the accretion of dust and gas in a local disk that has been subsumed by the planet from the circumsolar solar disk. On the other hand, the outer, mostly retrograde and highly eccentric and inclined satellites probably represent captured bodies that had formed elsewhere.

It is suggested here that the gap in the parent-satellite distance distribution in Figures 1 and 2 exists because the large satellites on each planet have, over time, gravitationally ‘cleared’ smaller bodies from their respective orbital regions around the planet, just as the planets have cleared other bodies from the region of their orbits around the Sun.

Specifically, the four Galilean satellites appear to have cleared all of the other putative satellites of Jupiter (red squares in Figures 1 and 2) that may have previously existed in their vicinity, Titan has cleared any potential rings between the E and Phoebe rings along with all of the satellites in its vicinity (except Hyperion) around Saturn (green triangles), Titania and Oberon have cleared any other satellites of Uranus (orange circles) from their orbital region, and Triton has cleared any other satellites of Neptune (plus signs).

It is noted that wide separations between satellite orbits do not necessarily imply that this has occurred through gravitational clearing. Indeed, a more massive, distant satellite may exert a much larger gravitational influence than the closer satellite and/or may be in a resonant orbital relationship with it. Also, outward migration of the satellites due to tidal forces may also generate increased separations and resonant relationships that mimic the effects of clearing.

Nonetheless, the prospect of orbital clearing as a factor in producing the non-random diameter-distance relationship in Figures 1 and 2 is explored in detail below.

3.2. Domain of influence and orbital stability of a celestial body

Two parameters have been used to describe the influential relationship between a large and a small celestial body in close proximity. The so-called ‘sphere of influence’ (SoI) has been used primarily for spacecraft mission planning. Its radius is provided by Equation (1):

(1) \begin{equation}\text{Radius of SoI} = a (m/M)^{2/5},\end{equation}

where a is the semimajor axis of the orbit of the smaller object around the larger body, and m and M are the masses of the smaller and larger objects, respectively. At a distance of a lunar spacecraft from the larger body (i.e., the Earth) less than the SoI, the satellite’s orbit can be approximated as motion around the Earth, perturbed by the smaller one (i.e., the Moon), whereas outside the SoI, the satellite’s motion is approximated as being around the Moon, perturbed by the Earth.

The closely related Hill (or Roche) sphereFootnote b Hamilton & Burns (Reference Hamilton and Burns1991) of a celestial body is the region in which it dominates the attraction of satellites; the outer shell of that region constitutes a zero-velocity surface. It is used to determine the distance from the larger body out to which the smaller object remains in a stable orbit. The radius of the Hill sphere (Hs) is given inEquation (2):

(2) \begin{equation}\text{Radius of Hs} = a (m/3M)^{1/3},\end{equation}

where the parameters have the same meaning as in Equation (1), and eccentricity of the orbit is negligible.

Itis not obvious which of the SoI or the Hs is more relevant in determining how closely objects can be placed around a star or planet, but it is a widely used criterion for determining orbital stability in two-body systems (Gladman Reference Gladman1993). For compact systems of more than two bodies—the situation under consideration in the current work—scaling through an exponent of 1/4 in Equation (2) has been shown to work better than 1/3 (Petit et al. Reference Petit, Pichierri, Davies and Johansen2020).

A body that forms or strays within the larger body’s SoI or Hs is unlikely to be captured permanently by that body (unless another influence comes into play, and it loses energy) but it may spend a few orbits around the larger body before returning to heliocentric orbit. Reid (Reference Reid2015) has argued that no ‘fourth order’ objects, namely, satellites of satellites, have been observed in the Solar System because tidal orbital evolution of these objects would have caused them to spiral into the parent (synchronous) satellite early in their history.

The SoI and Hs radii for the 20 largest satellites in the Solar System have been calculated using Equations (1) and (2) and are presented in Table 1, in descending order of satellite diameter. All of the satellites bar the last in this list, Proteus, have a spherical or near-spherical shape. All appear to have cleared their orbits, as judged from the diameter-distance distributions in Figures 1and 2.

Table 1. Mass, size, orbital dimensions and SoI radius of the 20 largest satellites in the Solar System.

^*The masses (kg) of the planets: Earth $5.972 \times 10^{24}$ , Jupiter $1.898 \times 10^{27}$ , Saturn $5.683 \times 10^{26}$ , Uranus $8.681 \times 10^{25}$ , Neptune $1.024 \times 10^{26}$ , Pluto $1.309 \times 10^{22}$ , Eris $1.66 \times 10^{22}$ .

#Calculated from $\mathrm{R}_{\mathrm{SOI}} \approx a(m/M)^{2/5}$ , where a is the semimajor axis of the satellite’s orbit around the planet, and m and M are the masses of the satellite and the planet, respectively.

##Calculated from $\mathrm{R}_{\mathrm{Hs}} \approx a(m/3M)^{1/3}$ , where a is the semimajor axis of the satellite’s orbit around the planet, and m and M are the masses of the satellite and the planet, respectively.

Raw data is from https://solarsystem.nasa.gov/moons/in-depth/, https://ssd.jpl.nasa.gov/?sat_elem, and https://ssd.jpl.nasa.gov/?sat_phys_par; formula from https:// en.wikipedia.org/wiki/Sphere_of_Influence_(astrodynamics)#Table_of_selected_SOI_radii

The SoI and Hs radii of the larger bodies (planets, dwarf planets and the Moon) and the Galilean satellites and their adjacent small moons are plotted in Figures 3(a) and (b) respectively.

Figure 3. Comparison of the radius of the Sphere of Influence (SoI) and Hill Sphere (Hs) for (a) the larger bodies in the Solar System (planets, dwarf planets and the Moon) and(b) the Galilean satellites of Jupiter and their adjacent smaller moons.

Figure 3 shows that the Hs radius is larger than the SoI radius by a factor of between 1.1 and 1.9 for the major planets and the Galilean satellites, but for the dwarf planets Pluto, Ceres and Eris, and for smaller satellites of Jupiter, Hs is between 2.6 and 6.0 times larger than SoI. Interestingly, for the systems that have been likened to binary planets, namely Pluto-Charon, Earth-Moon and Eris-Dysnomia, the Hs value for the satellite is smaller than the SoI, viz., 0.95, 0.93, and 0.80, respectively (Table 1).

With the exception of Mars, the upper limit of influenceFootnote c for the planets is larger than the largest observed planet-satellite apoapsis distances for each system (Figures 1, 2 and 3(a)) by a factor of between 1.50 (Neptune-Neso) and 3.91 (Earth-Moon); not unexpectedly for the unusual Mars system, the Mars-Deimos ratio is 46.0. This suggests that there is scope for many additional (non-Trojan) stable satellites to be discovered on allplanets.

Calculations of the Hs values for the Galilean satellites and their adjacent smaller moons (Figure 3(b)) are given particular attention in the discussion below in order to investigate their influence on these close neighbours in the context of large satellite ‘clearing’ capability.

In Figure 4(a), the closest distance of approach of adjacent planets and dwarf planets and in Figure 4(b) the closest approach of the Galilean satellites to their nearest neighbours have been expressed in multiples of the corresponding largest Hs value of the pair of bodies, calculated from a comparison of their orbital semimajor axes, assuming zero orbital eccentricity.

Figure 4. Closest separation distance between (a) the planets and dwarf planets, and (b) 17 of the largest satellites and their nearest neighbours, each expressed as a multiple of the largest of the two Hs radii of the relevant bodies.

The ratio of closest approach distance to the largest Hs radius for the adjacent planets and/or dwarf planet pairs (Figure 4(a)) varies roughly with the diameter of the body; it is largest for the dwarf planets (37–171), intermediate for the rocky planets(27–52), and smallest for the gas giants (4.6–20.6). While the variation in multiples of Hs radius is large, it surely indicates that the region around these bodies has been effectively cleared.

In terms of the major satellites (Figure 4(b)), the ratio of closest separation distance to Hs radius varies over a similar range to the planets and dwarf planets (with the minor exception of Ceres), namely, from 4.97 (Titan) to 57.2 (Iapetus), but appears to be unrelated to the diameter of the satellite. In both sets of bodies, the minimum separation distance is around 5 times the Hs radius. Thus, if the Hs ratio has any significance in the ‘clearing’ of the vicinity of celestial bodies (Figures 1 and 2), the mechanism may be similar for the satellites as it is for the planets.

This means that displacement/ejection of the smaller satellite body to another orbit well outside the clearing zone has occurred, or the smaller body has been captured into a resonant or other special orbital relationship, such as the Lagrangian points (as for the Trojans of Tethys and Dione) and the 3:4 resonant relationship between Hyperion and Titan.

3.3. Clearing the neighbourhood around a body’s orbit

Stern & Levison(Reference Stern and Levison2002), in the context of later discussion about the definition of a ‘planet’ during the 2006 IAU general assembly, developed a theoretical basis for determining if an object orbiting a star is likely to ‘clear its neighbouring region’ of planetesimals based on the object’s mass and its orbital dimensions. This was embodied in a dimensionless parameter $\Lambda$ (lambda), a measure of a body’s ability to scatter smaller masses out of its orbital region ‘over the age of the Universe’:

(3) \begin{equation}\Lambda = k m^2 / a^{3/2},\end{equation}

where m is the mass of the body relative to the Sun, a is the body’s semimajor axis, and k is a function of the orbital elements of the smaller body being scattered, including its orbital period, eccentricity, inclination, and the ratio of its semimajor axis to that of the larger body. Stern and Levison (hereafter SL) proposed that if $\Lambda > 1$ , the body will likely clear out smaller bodies in its orbital zone. They used this parameter to separate the gravitationally rounded, Sun-orbiting bodies into ‘Überplanets’, which are ‘dynamically important enough to have cleared their neighbouring planetesimals’, and ‘unterplanets’, the rest.

While the SL methodology is relatively crude and did not consider numerical orbital integrations, it was successful in that the Überplanets turned out to be the eight most massive IAU solar orbiters/planets, and the Moon (if it were in a heliocentric orbit) and the unterplanets were found to be similar (if not identical) to the IAU dwarf planets (Ceres, Pluto and an unidentified Kuiper Belt object).

In order to determine if $\Lambda$ can also be used to ascertain whether large satellites are capable of clearing their orbital domains, Equation (3) has been applied here to the major satellites of the gas giants. Adjustments to Equation (3) so that it bears on satellites rather than planets are that (i) m is the mass of each satellite relative to its specific parent planet (rather than the Sun), and (ii) a is the satellite’s semimajor axis (rather than the planet’s). To calibrate against the planets, Equation (3) has also been applied (in its original form) to the Sun-planet systems and incorporated with the other results in Figure 5.

Figure 5. Distribution of the Stern & Levison (Reference Stern and Levison2002) parameter $\Lambda$ , plotted as a function of the logarithm of the semimajor axis of the planets, dwarf planets and satellites of the gas giants. The zone between bodies classified by SL as Überplanets and unterplanets in the Solar System is designated by parallel dashed lines.

The distribution of $\Lambda$ and the semimajor axes for the planets and dwarf planets in the right half of Figure 5 is identical to that found by SL, although the magnitude of $\Lambda$ is different in an absolute sense since k has been set to unity for all calculations here. The two dashed lines in the Figure delineate the region between the IAU dwarf planets and IAU planets; it shows the same gap of 5 orders of magnitude in the value of $\Lambda$ between these classes of bodies as discovered by SL.

The $\Lambda$ values for the largest satellite systems of each of the four giant planets have been included in Figure 5 for ease of comparison with the results obtained for the planets and dwarf planets, despite the fact that k is different for each of these planetary systems due to differences in their orbital characteristics (viz., mass of the planet, relative and absolute orbital velocity, inclination, and ratio of semi-major axes). While internally consistent, without recalibration each of the planetary systems of satellites is displaced slightly along the y axis relative to each other and to the system of planets and dwarf planets.

By following the distribution of symbols for each planetary system in Figure 5, it is clear that there is a significant gap of 8 orders of magnitude between the $\Lambda$ values for the four Galilean and the smaller satellites of Jupiter, a smaller gap of 2-3 orders of magnitude between the 6 largest and the smaller satellites of Saturn, but no consistent break between satellites of Uranus and Neptune. Thus, while the gaps in satellite distributions for Jupiter and Saturn might be used under the SL methodology to unambiguously separate the clearing and non-clearing ones, the situation is not as clear for the small number of satellites around Uranus and Neptune.

Soter (Reference Soter2006) proposed an observational parameter, $\mu$ , toclassify bodies in nonresonant crossing orbits into planets and non-planets:

(4) \begin{equation}\mu = M / m,\end{equation}

where M is the mass of the candidate body and m is the mass of all of the other bodies that share its ‘orbital zone’. Soter proposed that if $\mu > 100$ then the body is classified as a planet. Since no such other bodies have been observed within at least 5 times the Hs radius (i.e., ‘orbital zone’) of any of the large satellites considered in the present study (presumably because they are very small or non-existent) the value of m is likely to be very small and $\mu$ would thus be much larger than 100 for all of the satellites. Therefore, the Soter parameter has not been explored further here.

Margot (Reference Margot2015) proposed another definition of planethood (albeit with specific reference to exoplanets) through a new parameter $\Pi$ (pi):

(5) \begin{equation}\Pi = k m / (M^{5/2} a^{9/8}),\end{equation}

where k is a constant chosen so that $\Pi > 1$ for a body that can clear its orbital zone, M is the mass of the parent star in solar masses, and m and a are the masses and semimajor axes, respectively, of the (exoplanet) body relative to the Earth.

Figure 6. Distribution of the Margot (Reference Margot2015) parameter $\Pi$ , calculated using Equation (5), plotted as a function of the logarithm of the semimajor axis of the planets, dwarf planets and satellites of the gas giants. The zone between bodies classified by the IAU as planets and dwarf planets in the Solar System is designated by dashed lines.

Figure 7. The relationship between the mass of a planet, dwarf planet or satellite and the logarithm of its orbital semimajor axis, as defined by Equation (8) in Margot (Reference Margot2015). The dashed line corresponds approximately to the position and slope of the line proposed by Margot (Reference Margot2015) to separate the eight planets from all other bodies in the Solar System.

Figure 6 shows the results of application of Equation (5) to the same set of planets and satellites as in Figure 5, with k and M set to 1, the latter because there is no need to normalise the mathematics to the Sun when exoplanets are not in the frame. The other adjustment made to Equation (5) in respect of the satellites is that their m and a values are normalised to the mass and semimajor axis of the specific parent planet (Jupiter, Saturn, Uranus or Neptune) rather than to the Earth.

Figure 6 shows an identical distribution of the planets and dwarf planets as does Margot’s $\Pi$ (and SL’s $\Lambda$ ) calculation, with a similar gap of 3 orders of magnitude between the $\Pi$ values for these two groups of bodies. The magnitude of the $\Pi$ values in Figure 6 are displaced relative to Margo’s $\Pi$ due to the setting of k and M to 1 in the current calculations.

Unlike SL’s parameter, Margot’s $\Pi$ reveals a much more distinct gap in the distribution of the larger and smaller gas giant satellites in the left-hand part of the Figure. Specifically, using Margot’s $\Pi$ , only the Galilean moons, along with Triton, Titan, Ariel, Umbriel, Titania, Oberon and Miranda lie above the upper ’clearing line’ for the planets (0.05, or about the same value as Mars) in the Figure. This distribution of 10 large satellites (plus Miranda) is closely similar to the set of 11 large satellites that were identified as potentially having cleared their orbital regions from the distributions plotted in Figures 1 and 2. Furthermore, lowering the clearing limit to around 0.002 (i.e., nearer to the midpoint of the gap between the planets and the dwarf planets) would bring the set of possible clearing bodies closer in alignment to the set identified by SL’s $\Lambda$ in Figure 5.

Equation (8) of Margot (Reference Margot2015) seeks to predict the mass of a body above which it becomes capable of clearing its orbit, through a simple plot of the mass of the body versus its semimajor axis. In this analysis Margot drew a line separating those bodies in the Sun-planet system that had cleared their orbits and those that had not, with a slope dependent on the clearing time, planet/star mass ratio, and the Hill Radius of the body.

Margot’s large-body analysis has been reproduced in Figure 7, with the additional inclusion of satellite data. The dashed line is drawn in a position and with a slope analogous to that of Margot (Reference Margot2015). When projected down in mass to the area occupied by the planetary satellites, this line differentiates between the large and small satellites in a similar way to the clearing lines in Figures 5 and 6. Once again, the satellites proposed in Figures 1 and 2 to have cleared their orbital domains all lie above the Margot line in Figure 7.

The figure shows that the mass required for clearing an orbital region reduces with the size of the semimajor axis of that orbit. It suggests that the minimum mass for clearing decreases from around $1.8\times 10^{21}\,\mathrm{kg}$ for $1\,470\,\mathrm{km}$ diameter Iapetus on Saturn, with a semimajor axis of $3.6 \times 10^6\,\mathrm{km}$ , to around $5.0 \times 10^{19}\,\mathrm{kg}$ for $420\,\mathrm{km}$ diameter Proteus on Neptune, with a semimajor axis of $1.2 \times 10^5\,\mathrm{km}$ .