Landau Tidal Damping and Major-Body Clustering in Solar and Extrasolar Subsystems

Abstract

Major (exo)planetary and satellite bodies seem to concentrate at intermediate areas of the radial distributions of all the objects orbiting in each (sub)system. We show that angular-momentum transport during secular evolution of (exo)planets and satellites necessarily results in the observed intermediate accumulation of the massive objects. We quantify the ‘middle’ as the mean of mean motions (orbital angular velocities) when three or more massive objects are involved. Radial evolution of the orbits is expected to be halted when the survivors settle near mean-motion resonances and angular-momentum transfer between them ceases (gravitational Landau damping). This dynamical behavior is opposite in direction to what has been theorized for viscous and magnetized accretion disks, in which gas spreads out and away from either side of any conceivable intermediate area. We present angular momentum transfer calculations in few-body systems, and we also calculate the tidal dissipation timescales and the physical properties of the mean tidal field in planetary and satellite (sub)systems.

1. Introduction

An inspection of planetary and satellite orbital data in the Solar System (https://ssd.jpl.nasa.gov, https://solarsystem.nasa.gov, accessed on 16 April 2024) reveals that major objects seem to cluster at intermediate areas of the radial distributions of orbiting bodies, and only smaller objects are found in the inner and the outer regions of these subsystems. The same arrangement of massive objects is usually seen in multiplanet extrasolar systems as well. Keeping in mind that there may be more undetected planets farther out in these systems, some examples presently are HD 10180 [1], Kepler-80 and 90 [2], TRAPPIST-1 [3,4], HR 8832 [5,6,7], K2-138 [8,9], Kepler-11 [10], and even the four-planet systems of Kepler-223 [11] and GJ 876 [12,13]. Despite being a clue pertaining to the processes of massive planet and large satellite formation and evolution, this conspicuous property has not been discussed in the past, and there have been no ideas about how we could possibly exploit it to learn from it.

Our approach to the problem has been single-minded from the outset. It was apparent to us that such large bodies have moved toward one another during early evolution, perhaps as soon as a few large solid cores emerged in these subsystems and the accretion disks dissipated away. In such a case, there must exist a generic physical mechanism (a global tidal field) that drives angular-momentum redistribution, but, eventually, further migration is hindered when this mechanism ceases to operate. In this work, we formulate such a secular mechanism (the gravitational Landau damping of the tidal field) that relies on first principles and requires no additional conditions in order to operate. Some related calculations, commonly involving continuous fluids or collisionless systems with at least ${10}^{10}$ particles, have been carried out by other researchers in the past [14,15,16,17,18,19,20]. The differences that we point out below concern the details of evolution of just a few (4–7) major bodies and the physical interpretation of the results.

In Section 2, we describe the dynamical evolution of two interacting Keplerian fluid elements through nonequilibrium states that leads to a runaway dynamical instability. This analysis is applicable to magnetized accretion disks, but not to planets or satellites in which the integral of circulation is not conserved (not even approximately—these systems are topologically not simply connected). In Section 3, we describe the secular evolution of large individual gravitating bodies in Keplerian orbits around a central mass and under the influence of tidal dissipation which leads to body clustering. In Section 4, we discuss our results in the context of planet and satellite evolution.

Quite a few technical details are deferred to in three self-contained appendices:

• In Appendix A, we describe few-body systems evolving by exchanging angular momentum and lowering their total mechanical energies.
• In Appendix B, we formulate a self-consistent calculation of the characteristic tidal dissipation time ${\tau }_{\mathrm{dis}}$ and the corresponding radial velocity fluctuations $v_{\mathrm{dis}}$ in such systems.
• In Appendix C, we analyze gravitational Landau damping of the tidal field in few-body systems, a unique mechanism responsible for settling of the bodies near mean-motion resonances over times comparable to ${\tau }_{\mathrm{dis}}$—where they no longer exchange substantial amounts of angular momentum, and so they send the global tidal field to oblivion.

The contents of the appendices are also discussed in context in Section 4.1.

2. Dynamical Evolution of Interacting Keplerian Fluid Elements

Balbus and Hawley [14] introduced a mechanical analog of the magnetorotational instability (MRI) in gaseous accretion disks, two mass elements $m_1$ and $m_2$ in circular Keplerian orbits around a central mass $M\gg m_1,m_2$ with radii $r_1$ and $r_2>r_1$, respectively. The mass elements are connected by a weak spring with constant k (representing a magnetic-field line) whose role is to allow for angular momentum transfer between the elements. When perturbed under the constraint of constant total angular momentum (constant circulation would be more precise, although the two integrals of motion are equivalent in axisymmetric fluid systems), this model behaves just like gaseous accretion disks under the influence of viscosity [16], except that the instability is now dynamical: the masses move apart and their displacements reduce the total free energy of the system [21], leading to a runaway in the separation of the two masses [14,22,23,24].

We use a free-energy variational formalism [21] to describe the evolution of this system as perturbations take it out of equilibrium: if a perturbation lowers the free energy ($\Delta E<0$) while preserving the total angular momentum ($\Delta L=0$), the system will transition to a new nonequilibrium state of lower energy; whereas if $\Delta E>0$, the system will just oscillate about the initial equilibrium state characterized by total energy $E=E_1+E_2$ and total angular momentum $L=L_1+L_2$. We assume that the initial equilibrium orbits are perturbed by small displacements $\Delta r_1\ll r_1$ and $\Delta r_2\ll r_2$. Then, the conservation of total angular momentum relates the displacements to first order by the equation

$$m_1{v}_1\Delta r_1+m_2{v}_2\Delta r_2=0,$$

(1)

where $v_1$ and $v_2$ are the equilibium azimuthal velocities, and the change in free energy to first order is found to be

$$\Delta E=L_1(n_1-n_2)\frac{\Delta r_1}{2r_1},$$

(2)

where $L_1\equiv m_1{r}_1{v}_1$, and $n_1$ and $n_2<n_1$ are the mean motions (orbital angular velocities) of the masses in their equilibrium state. The change in potential energy of the spring, $k{(\Delta r_2-\Delta r_1)}^2/2$, is of second order and is omitted from Equation (2). It is now apparent that for $\Delta r_1<0$, then $\Delta E<0$ and $\Delta r_2>0$. The masses move apart and the resulting nonequilibrium configuration is unstable to continuing disengagement that will reduce further the free energy of the system.

The above dynamical instability (a mechanical analog of the MRI) does not operate in planetary and satellite systems. It is strictly applicable to perfect fluids in which circulation and angular momentum are both conserved [21]. Conservation of circulation is implicit in the above model; it can be readily seen in Equation (1) assuming that the mass elements are axisymmetric rings with equal masses, in which case the equation takes the equivalent form

$$v_1\Delta r_1+v_2\Delta r_2=0,$$

(3)

to first order in the displacements.

In viscous unmagnetized disks, dissipative stresses destroy circulation slowly and the instability is then secular [16]. In stellar and particle systems, there is no conservation law of circulation and Equation (3) is invalid, even in approximate form, because all the elements of the stress tensor introduce gradients of comparable magnitude into the Jeans equations of motion [21,25,26]. Therefore, the evolution of multiple planetary and satellite bodies requires a different mathematical approach, though still constrained by the applicable conservation laws of energy and angular momentum.

3. Secular Evolution of Interacting Planets and Satellites

3.1. Quasistatic Equilibria

Ostriker and Gunn [17] studied the secular evolution of a dynamically stable, uniformly-rotating body subject to angular momentum and energy losses due to emission of multipolar radiation. Evolution takes place slowly over timescales much longer than the dynamical time (the rotation period) of the object. In this model, the body is thought of as transitioning between quasistatic equilibrium states (the Dedekind ellipsoids [27]) in which it maintains its uniform rotation albeit with a slowly changing angular velocity $\mathsf{\Omega }$. Here, ‘slowly’ is quantified by the condition that

$$\left|\frac{d\mathsf{\Omega }}{dt}\right|\ll {\mathsf{\Omega }}^2.$$

(4)

Under a series of assumptions, the strongest of which is inequality (4), Ostriker and Gunn [17] proved that the losses in angular momentum L and rotational kinetic energy E are related by the equation

$$\frac{dE}{dt}=\mathsf{\Omega }\frac{dL}{dt},$$

(5)

where the time derivatives are both implicitly negative. The use of E for rotational kinetic energy (their Equation (7)) has caused some indiscretions in the literature. For example, Page and Thorne [18] call E the ‘energy-at-infinity’ (which is kinetic after all) and Equation (5) ‘universal’ despite having derived it under their assumption iv(a), which is essentially equivalent to condition (4); whereas Papaloizou [19] calls E the ‘orbital energy’ of a planet in the context of the same quasistatic approximation.

Below, we also use Equation (5) to follow the secular evolution of planets and satellites losing kinetic energy slowly due to the action of dissipative processes induced by a global tidal field. First, we revisit the approach of Papaloizou [19], whose calculation is correct but his conclusion is wrong. Then, we formulate the same problem as a variation of the free energy of the system [21] undergoing quasistatic out-of-equilibrium evolution away from its initial equilibrium state.

3.2. Papaloizou Approach

We consider two gravitating bodies with masses $m_1$ and $m_2$ orbiting around a central mass $M\gg m_1,m_2$ in nearly circular Keplerian orbits with radii $r_1$ and $r_2>r_1$, respectively. We assume that tides due to M during orbit circularization are dissipated in the interiors of the bodies, causing small amounts of kinetic energy to be converted to heat H. The slow rate of dissipation is given by

$$\mathcal{L}=dH/dt>0.$$

(6)

Here, ‘slow’ is defined by inequality (4) and by the condition that

$$H\ll T,$$

(7)

where T is the total orbital kinetic energy of the bodies. Then, the evolution of the system is described by a sequence of quasistatic equilibrium states that are accessible to the bodies because Equation (6) together with energy conservation guarantee that the total mechanical energy E of the bodies will decrease in time ($dE/dt<0$).

The mechanical energy and angular momentum contents of each body are related by

$$E_i=-\frac{1}{2}{n}_i{L}_i,$$

(8)

where $i=1,2$ and $n_i$ is the mean motion of body $m_i$. Since $r_2>r_1$, then $n_2<n_1$ for the Keplerian orbits. Under the quasistatic assumption (4), a relation analogous to Equation (5) is also applicable here, in the form

$$\frac{d{E}_i}{dt}=-\frac{1}{2}{n}_i\frac{d{L}_i}{dt}.$$

(9)

The factor of $-1/2$ appears because $E_i$ represents the mechanical energy of each body, which is implicitly negative ($E_i\equiv -GM{m}_i/\left(2r_i\right)$, where G is the Newtonian gravitational constant). The negative sign cannot be absorbed in Equations (8) and (9) because, unlike $dL/dt<0$ in Equation (5) above, here, the individual terms $d{L}_1/dt$ and $d{L}_2/dt$ have opposite signs.

Conservation of total angular momentum $L=L_1+L_2$ is expressed by the equation

$$\frac{d}{dt}\left(L_1+L_2\right)=0,$$

(10)

and total energy conservation for the system gives

$$\frac{d}{dt}\left(E_1+E_2\right)=-\frac{dH}{dt}=-\mathcal{L}<0.$$

(11)

Using Equation (9), we rewrite Equation (10) in the form

$$\frac{1}{n_1}\frac{d{E}_1}{dt}+\frac{1}{n_2}\frac{d{E}_2}{dt}=0.$$

(12)

Thus, after considerable deliberations of the details, we have arrived at the equations adopted by Papaloizou [19].

It is obvious from Equations (11) and (12) that, as the system evolves quasistatically, the mechanical energy of one body will increase and that of the other body will decrease; but the overall change in $(E_1+E_2)$ will be a decrease by an amount of $dH$, allowing for the system to proceed to a neighboring quasi-equilibrium state. In the absence of a model for $dH/dt$, it is not prudent to solve these equations for the energy rates in order to deduce the details of the evolution. It is more sensible to examine the changes in angular momentum of the bodies: Combining Equations (9)–(11), we find that

$$-\frac{d{L}_2}{dt}=\frac{d{L}_1}{dt}=\frac{2\mathcal{L}}{n_1-n_2}>0,$$

(13)

where $\mathcal{L}>0$ and $n_1>n_2$. We see now that the inner body 1 will gain angular momentum and will move outward, while the outer body 2 will lose angular momentum and will move inward. Overall, the two bodies will converge toward an intermediate orbit in which they would both share the total angular momentum equally.

Surprisingly, convergence toward the intermediate orbit with the average angular momentum $\overline{L}$ does not occur in three or more orbiting bodies (Appendix A), where a new ‘critical orbit’ emerges, characterized by a mean motion $\overline{n}$, the average of the initial mean motions of the bodies. But in these systems, convergence toward this critical orbit will not continue unscathed, if two-body encounters set in and upset the quasistatic evolution. For the same reason, this orbit is, in principle, secularly unstable, but a body placed on it will remain in place for a long time, provided that another orbiting body does not come close. These results are established in Appendix A, and their significance and repercussions are discussed in Section 4.

3.3. Free-Energy Variational Approach

We briefly formulate the problem studied in Section 3.2 as a variation of the free energy of the system of two bodies with masses $m_i$ ($i=1,2$) orbiting around a central mass $M\gg m_i$ and stepping out of equilibrium and into a new state while still obeying conditions (4), (7), and (9). The two bodies can move on to such a (generally nonequilibrium) state only if this state is characterized by lower free energy ($\Delta E<0$) and the same total angular momentum ($\Delta L=0$). The total mechanical energy $(E_1+E_2)$ plays the role of the free energy function [21] here; thus, we have

$$\Delta \left(E_1+E_2\right)<0,$$

(14)

and

$$\Delta \left(L_1+L_2\right)=0.$$

(15)

Combining these two relations with Equation (9) in the form $\Delta E_i=-(1/2)n_i\Delta L_i$, we find that

$$\left(n_2-n_1\right)\Delta L_2=\left(n_1-n_2\right)\Delta L_1>0.$$

(16)

For $n_1>n_2$ (implying that the initial Keplerian radii obey $r_1<r_2$), we find that $\Delta L_1>0$ and $\Delta L_2<0$, respectively. Thus, in order for the system to begin its search for a new equilibrium state of lower free energy, the inner body $m_1$ will move outward and the outer body $m_2$ will move inward.

4. Summary and Discussion

4.1. Summary

We have used the conservation laws of energy and angular momentum to describe and contrast the dynamical evolution of two interacting mass elements in a gaseous accretion disk and the secular evolution of massive planets and large satellites. Both types of subsystems were assumed to exhibit Keplerian orbital profiles around a dominant central mass and to exchange angular momentum via weak tidal torques. But evolution takes different paths in these two cases, and the reason is the (non)conservation of circulation. In perfect-fluid disks (Section 2), circulation is conserved and the mechanical analog of the MRI suffers a dynamical instability, as was first described by Balbus and Hawley [14]; whereas in (extra)solar multi-body subsystems (Section 3), there is no analogous conservation law, and dissipative evolution proceeds secularly via a sequence of quasistatic equilibrium configurations [17] or via nonequilibrium states, both of which progressively lower the total mechanical energy of the (sub)system.

Extending the analytical work of Papaloizou [19] to more than two orbiting bodies, we have shown in Appendix A that tidal dissipation induced by the central mass leads to clustering of multi-body systems generally toward the mean $\overline{n}$ of their initial mean motions $n_i$, where $i=1,2,\cdots ,N$ and $N\ge 3$. (In contrast, two orbiting bodies converge toward an intermediate orbit with angular momentum $\overline{L}=(L_1+L_2)/2$ (Section 3.2)). Although secularly unstable, this critical orbit may host a massive body for a long time, at least comparable to the dissipation time ${\tau }_{\mathrm{dis}}$ that characterizes this part of the evolution of the system (${\tau }_{\mathrm{dis}}$ is quantified in Appendix B). On the other hand, a close encounter with another body can clear out the critical orbit, if the interaction between the two bodies continues unimpeded for a long enough time.

Convergence of major bodies toward an intermediate region of the radial distribution of orbiting bodies should not come as a surprise, as it is observed in quite a few solar subsystems (gaseous giants, Galilean moons, and Saturnian and Uranian moons) and many exoplanetary systems (Section 1). These (sub)systems show an unmistaken clustering of several (4–7) massive bodies at intermediate orbital locations that appear to be gathered around the critical orbit with mean motion $\overline{n}$. (Neptune’s primordial satellite system was obliterated by the arrival and retrograde capture of Triton, so it is not structurally comparable to the satellite systems of the other gaseous giants).

The question then is, where and how does such radial convergence of orbiting bodies stop? After all, the observed massive planets and satellites seem to be currently on very long-lived, if not secularly stable, orbits, with no pair evolving toward an in-between orbit. Therefore, the clustering process must be quelled somehow before the objects begin interacting strongly via close paired encounters. The seminal paper of Goldreich [15] long ago provided a substantial part of the answer: Goldreich [15] showed that several “special cases of commensurable mean motions [of satellites] are not disrupted by tidal forces”. This means that when some of the more massive satellites of the gaseous giants reach near mean-motion resonances (MMRs), their angular momenta do not vary tidally any more; thus, the satellites maintain their orbital elements in long-lasting dynamical configurations, and they no longer contribute to the mean tidal field. In the process, the tidal field is weakened, and it is damped out completely when all massive bodies reside in MMRs. We have analyzed in Appendix C this type of gravitational Landau damping of the tidal field generated by few bodies.

4.2. Global Mean-Motion Resonances

The most massive body must play an important role in the above process because it is the one that evolves tidally slower than all the other bodies; so, it must be the body that lays out the resonant structure of the global tidal field for the entire (sub)system. When other massive bodies reach nearby global MMRs, their further evolution is impeded because the most massive body does not affect them tidally any longer; they also refrain from interacting with smaller orbiting bodies in the system. In this setting, the tidal field will thus be severely damped and the remaining low-mass objects trying slowly to converge will also be hampered, either because they encounter global MMRs, or they are simply too far away from the resonating massive bodies. In the end, the entire system will appear to be relaxed (no more substantial imbalances from exchanges of angular momentum), with all of its members lying in or near global MMRs and the mean tidal field erased since the major bodies no longer contribute to it. At present, this is what is actually observed in all (exo)planetary and satellite subsystems.

At this point, we should clarify what we perceive differently in reference to the volumes of work carried out about (mostly local) MMRs (page https://en.wikipedia.org/wiki/Orbital_resonance contains a comprehensive empirical summary of orbital MMRs along with a listing of hyperlinks to ∼100 professional citations) to date [8,10,12,13,15,28,29,30,31,32,33,34]: We argue that multi-body resonances are not a local phenomenon; ‘principal MMRs’ (1:k and k:1, where $k=1,2,3,\cdots$) are global in each system, and their locations are set by the most massive object that used to dominate the mean tidal field affecting the entire system; and secondary global MMRs of the form p:q ($p,q\ne 1$) are also bound to lie at in-between locations. In such a global layout, it is not helpful to focus on the relative deviations of orbital elements from exact nearby MMRs and devise arbitrary thresholds for objects to be, or not to be, locked in resonance. Though, unfortunately, we recognize this to be the current state of affairs in studies of phase angles of local MMRs between near-neighbors: for instance, the Earth is not thought to be in the 1:12 global MMR of Jupiter because its orbital period is 4.2 days longer than the exact resonant value of 361.05 d; and its phase angle would have to circulate slowly relative to the phase of Jupiter, so the same pattern would only repeat once every 87 years (see section “Coincidental Near MMRs” in https://en.wikipedia.org/wiki/Orbital_resonance for the same argument). This is not the right way of thinking about global resonances in the mean tidal field of our solar system (that is severely damped presently). We defer further discussion of this rather complicated issue to Appendix C.

4.3. The Critical Orbit in Solar Subsystems

The main result of this work has ramifications beyond the particular systems that we study. The orbits of the planets and satellites all have Keplerian radial profiles. The Keplerian profile is just a special case of a power law, a profile with no critical or inflection points, a property that makes it relatively simple, but also totally featureless.

But now, the dynamics of multiple bodies evolving by applying torques and exchanging angular momentum has given us a critical point in this profile, the mean $\overline{n}$ of the initial mean motions $n_i$, or, equivalently, the harmonic mean $\overline{P}$ of the orbital periods $P_i$ ($i=1,2,\cdots ,N$, where we take $N\ge 3$), viz.,

$$\overline{P}=\frac{N}{{\displaystyle \sum _{i=1}^N}{\left(P_i\right)}^{-1}}(N\ge 3).$$

(17)

Given $\overline{P}$, the critical orbital radius can be determined from Kepler’s third law. We note, however, that not many bodies are expected to occupy the critical orbits in their subsystems because all bodies may have a priori circularized their orbits at or near MMRs—unless, of course, the critical orbit coincides with an MMR, in which case the chances of finding a body there improve considerably (see Section 4.4 below).

Our planetary system and Jupiter’s satellite subsystem each contain $N=4$ dominant adjacent orbiting bodies, the gaseous giant planets and the Galilean moons, respectively. For the gaseous giants, we find that

$$\overline{P}=29.36\mathrm{y}(\mathrm{whereas}{P}_{\mathrm{Sa}}=29.46\mathrm{y}),$$

so Saturn has settled just wide of the critical orbit as we see it presently. For the Galilean moons, we find that at present,

$$\overline{P}=3.82\mathrm{d}(\mathrm{whereas}{P}_{\mathrm{Eu}}=3.55\mathrm{d}),$$

so Europa was trapped into the renowned Laplace resonance (LR) and could not expand its orbit farther out. We did not include inner low-mass bodies in these estimates for an obvious reason; their fates were fully determined by weak tidal forces exerted on them by the distant massive bodies, so they can be viewed as passive receivers of tiny amounts of angular momentum having slowly worked their way outward and toward the common goal. The Earth, in particular, may have taken angular momentum from nearby Mars, preventing the outward movement of this tiny planet.

For the Earth, it is interesting to examine where our planet finally settled at the end of the orbital evolution of the gaseous giants: our planet is currently orbiting just wide of the 1:12 principal MMR of Jupiter (as already mentioned, its orbital period is only 4.2 d longer). It may not be surprising that the planet did not settle at the center of the 1:12 MMR. During secular evolution, it was only gaining angular momentum, working its way outward toward the common goal. Such slightly wider orbits are observed in many exoplanets as well [10,28]. Those inner ones with orbital periods shorter than $\overline{P}$ may be understood along the same line of reasoning, as having moved outward during evolution (but see also Refs. [35,36] discussing this issue).

4.4. The Critical Orbit in Exoplanets

In extrasolar systems, K2-138 [8,9] presents a transparent example of a planet on a critical orbit. For the six planets known in this system, we find that

$$\overline{P}=5.39\mathrm{d}(\mathrm{whereas}{P}_d=5.40\mathrm{d}),$$

so planet d is effectively occupying the critical orbit. All planets are near global MMRs as determined from the orbital period of the largest planet e. In order of increasing orbital periods, these are 2:7, 3:7, 2:3 1:1, 3:2, 5:1, for planets b–g, respectively. In planets b–f, all adjacent pairs have local period ratios $P_{i+1}/P_i\simeq$ 3/2 [8]; and the outermost planet g resides in a higher-order harmonic ratio, i.e., $P_g/P_e\simeq {(3/2)}^4\simeq 5$. The resonant chain is global, though not fully packed. If it were fully packed, then no planet would occupy the hypothetical ‘8-planet’ critical orbit ($\overline{P}=6.66\mathrm{d}$).

Another example with the critical orbit being occupied is the TRAPPIST-1 system with seven planets in a very compact configuration ($r_{\max }=0.062$ AU; [3,4]). We find that

$$\overline{P}=P_d=4.05\mathrm{d},$$

so planet d lies on the critical orbit. All planets are near global MMRs, as determined from the orbital period of the largest planet g. In order of increasing orbital periods, these are 1:8, 1:5, 1:3, 1:2, 3:4, 1:1, 3:2, for planets b-h, respectively.