Solar System Dynamics: Regular and Chaotic Motion.
Solar System Dynamics: Regular and Chaotic Motion
Jack J. Lissauer, NASA Ames Research CenterMoffett Field, California
Carl D. Murray, Queen Mary, University of LondonLondon, U.K.
1. Introduction: Keplerian Motion
The study of the motion of celestial bodies within our solar system has played a key role in the broader development of classical mechanics. In 1687, Isaac Newton published his Principia, in which he presented a unified theory of the motion of bodies in the heavens and on the Earth. Newtonian physics has proven to provide a remarkably good description of a multitude of phenomena on a wide range of length scales. Many of the mathematical tools developed over the centuries to analyze planetary motions in the Newtonian framework have found applications for terrestrial phenomena. The concept of deterministic chaos, now known to play a major role in weather patterns on the Earth, was first conceived in connection with planetary motions (by Poincare, in the late 19th century). Deviations of the orbit of Uranus from that predicted by Newton's Laws led to the discovery of the planet Neptune. In contrast, the first major success of Einstein's general theory of relativity was to explain deviations of Mercury's orbit that could not be accounted for by Newtonian physics. But general relativistic corrections to planetary motions are quite small, so this article concentrates on the rich and varied effects of Newtonian gravitation, together with briefer descriptions of non-gravitational forces that affect the motions of some objects in the solar system.
Newton showed that the motion of two spherically symmetric bodies resulting from their mutual gravitational attraction is described by simple conic sections (see Section 2.4). However, the introduction of additional gravitating bodies produces a rich variety of dynamical phenomena, even though the basic interactions between pairs of objects can be straightforwardly described. Even few-body systems governed by apparently simple nonlinear interactions can display remarkably complex behavior, which has come to be known collectively as chaos. On sufficiently long timescales, the apparently regular orbital motion of many bodies in the solar system can exhibit symptoms of this chaotic behavior.
An object in the solar system exhibits chaotic behavior in its orbit or rotation if the motion is sensitively dependent on the starting conditions, such that small changes in its initial state produce different final states. Examples of chaotic motion in the solar system include the rotation of the Saturnian satellite Hyperion, the orbital evolution of numerous asteroids and comets, and the orbit of Pluto. Numerical investigations suggest that the motion of the planetary system as a whole is chaotic, although there are no signs of any gross instability in the orbits of the planets. Chaotic motion has probably played an important role in determining the dynamical structure of the solar system.
In this article the basic orbital properties of solar system objects (planets, moons, minor bodies, and dust) and their mutual interactions are described. Several examples are provided of important dynamical processes that occur in the solar system and groundwork is laid for describing some of the phenomena that are discussed in more detail in other articles of this encyclopedia.
1.1 Kepler's Laws of Planetary Motion
By analyzing Tycho Brahe's careful observations of the orbits of the planets, Johannes Kepler deduced the following three laws of planetary motion:
1. All planets move along elliptical paths with the Sun at one focus. The heliocentric distance r (i.e., the planet's distance from the Sun) can be expressed as
r = , (1)
with a the semimajor axis (average of the minimum and maximum heliocentric distances) and e (the eccentricity of the orbit) [?] (1 [?] b [sup.2]/a [sup.2])[sup.1/2], where 2b is the minor axis of an ellipse. The true anomaly, f, is the angle between the planet's perihelion (closest heliocentric distance) and its instantaneous position (Fig. 1).
FIGURE 1 Geometry of an elliptical orbit. The Sun is at one focus and the vector r denotes the instantaneous heliocentric location of the planet (i.e., r is the planet's distance from the Sun). a is the semimajor axis (average heliocentric distance), and b is the semiminor axis of the ellipse. The true anomaly, f, is the angle between the planet's perihelion (closest heliocentric distance) and its instantaneous position.
2. A line connecting a planet and the Sun sweeps out equal areas DA in equal periods of time Dt:
= constant. (2)
Note that the value of this constant differs from one planet to the next.
3. The square of a planet's orbital period P about the Sun (in years) is equal to the cube of its semimajor axis a (in AU):
P [sup.2] = a [sup.3]. (3)
1.2 EllipticaL Motion, Orbital Elements, and the Orbit in Space
The Sun contains more than 99.8% of the mass of the known solar system. The gravitational force exerted by a body is proportional to its mass (Eq. 5), so to an excellent first approximation the motion of the planets and many other bodies can be regarded as being solely due to the influence of a fixed central pointlike mass. For objects like the planets, which are bound to the Sun and hence cannot go arbitrarily far from the central mass, the general solution for the orbit is the ellipse described by Eq. (1). The orbital plane, although fixed in space, can be arbitrarily oriented with respect to whatever reference plane is chosen (such as Earth's orbital plane about the Sun, which is called the ecliptic, or the equator of the primary). The inclination, i, of the orbital plane is the angle between the reference plane and the orbital plane and can range from 0 to 180deg. Conventionally, bodies orbiting in a direct sense, with orbital angular momentum vectors within 90deg of the direction of the Earth's orbital angular momentum (or the rotational angular momentum of the primary), are defined to have inclinations from 0deg to 90deg and are said to be on prograde orbits. Bodies traveling in the opposite direction are defined to have inclinations from 90deg to 180deg and are said to be on retrograde orbits. The two planes intersect in a line called the line of nodes and the orbit pierces the reference plane at two locations--one as the body passes upward through the plane (the ascending node) and one as it descends (the descending node). A fixed direction in the reference plane is chosen and the angle to the direction of the orbit's ascending node is called the longitude of the ascending node, O. Finally, the angle between the line to the ascending node and the line to the direction of periapse (perihelion for orbits about the Sun, perigee for orbits about Earth) is called the argument of periapse o. An additional angle, the longitude of periapse p = o + O is sometimes used in place of o. The six orbital elements a, e, i, O, o and f uniquely specify the location of the object in space (Fig. 2). The first
FIGURE 2 Geometry of an orbit in three dimensions. The Sun is at one focus of the ellipse (O) and the planet is instantaneously at location R. The location of the perihelion of the orbit is P. The intersection of the orbital plane (X [?] Y) and the reference plane is along the line ON (where N is the ascending node). The various angles shown are described in the text. The mean anomaly M is an angle proportional to the area OPR swept out by the radius vector OR (Kepler's second law).
three quantities (a, e, and i) are often referred to as the principal orbital elements, as they describe the orbit's size, shape, and tilt, respectively.
2. The Two-body Problem
In this section the general solution to the problem of the motion of two otherwise isolated objects in which the only force acting on each body is the mutual gravitational interaction is discussed.
2.1 Newton's Laws of Motion and the Universal Law of Gravitation
Although Kepler's laws were originally found from careful observation of planetary motion, they were subsequently shown to be derivable from Newton's laws of motion together with his universal law of gravity. Consider a body of mass m [sub.1] at instantaneous location r [sub.1] with instantaneous velocity v [sub.1] [?] d r [sub.1]/dt and hence momentum p [sub.1] [?] m [sub.1] v [sub.1]. The acceleration d v [sub.1]/dt produced by a net force F [sub.1] is given by Newton's second law of motion:
F [sub.1] = . (4)
Newton's universal law of gravity states that a second body of mass m [sub.2] at position r [sub.2] exerts an attractive force on the first body given by
F [sub.1] = [?]r [sub.12] = [?]r̂ [sub.12], (5)
where r [sub.12] [?] r [sub.1] [?] r [sub.2] is the location of particle 1 with respect to particle 2, r̂ [sub.12] is the unit vector in the direction of r [sub.12], and G is the gravitational constant. Newton's third law states that for every action there is an equal and opposite reaction; thus, the force on each object of a pair is equal in magnitude but opposite in direction. These facts are used to reduce the two-body problem to an equivalent one-body case in the next subsection.
2.2 Reduction to the One-body Case
From the foregoing discussion of Newton's laws, and the two-body problem the force exerted by body 1 on body 2 is
= F [sub.2] = [?]F [sub.1] = r [sub.12] = r̂ [sub.12], (6)
Thus, from Eqs. (4) and (6)
= F [sub.1] + F [sub.2] = 0. (7)
This is of course a statement that the total linear momentum of the system is conserved, which means that the center of mass of the system moves with constant velocity.
Multiplying Eq. (6) by m [sub.1] and Eq. (5) by m [sub.2] and subtracting, the equation for the relative motion of the bodies can be cast in the form
m[sub.r] = m[sub.r] = [?]r [sub.12], (8)
where m[sub.r] [?] m [sub.1] m [sub.2]/(m[sub.l] + m [sub.2]) is called the reduced mass and M [?] m [sub.1] + m [sub.2] is the total mass. Thus, the relative motion is completely equivalent to that of a particle of reduced mass m[sub.r] orbiting a fixed central mass M. For known masses, specifying the elements of the relative orbit and the positions and velocities of the center of mass is completely equivalent to specifying the positions and velocities of both bodies. A detailed solution of the equation of motion (8) is discussed in any elementary text on orbital mechanics and in most general classical mechanics books. In the remainder of Section II, a few key results are given.
2.3 Energy, Circular Velocity, and Escape Velocity
The centripetal force necessary to keep an object of mass m[sub.r] in a circular orbit of radius r with speed v [sub.c] is m[sub.r] v [sub.c] [sup.2]/r. Equating this to the gravitational force exerted by the central body of mass M, the circular velocity is
v [sub.c] = . (9)
Thus the orbital period (the time to move once around the circle) is
P = 2pr/v [sub.c] = 2p. (10)
The total (kinetic plus potential) energy E of the system is a conserved quantity:
E = T + V = 1/2m[sub.r] v [sup.2] [?] , (11)
where the first term on the right is the kinetic energy of the system, T, and the second term is the potential energy of the system, V. If E < 0, the absolute value of the potential energy of the system is larger than its kinetic energy, and the system is bound. The body will orbit the central mass on an elliptical path. If E > 0, the kinetic energy is larger than the absolute value of the potential energy, and the system is unbound. The relative orbit is then described mathematically as a hyperbola. If E = 0, the kinetic and potential energies are equal in magnitude, and the relative orbit is a parabola. By setting the total energy equal to zero, the escape velocity at any separation can be calculated:
v [sub.e] = = [?]2v [sub.c]. (12)
For circular orbits it is easy to show [using Eqs. (9) and (11)] that both the kinetic energy and the total energy of the system are equal in magnitude to half the potential energy:
T = [?]1/2V, (13)
E = [?]. (14)
For an elliptical orbit, Eq. (14) holds if the radius r is replaced by the semimajor axis a:
E = [?]. (15)
Similarly, for an elliptical orbit, Eq. (10) becomes Newton's generalization of Kepler's third law:
P [sup.2] = . (16)
It can be shown that Kepler's second law follows immediately from the conservation of angular momentum, L:
= = 0. (17)
2.4 Orbital Elements: Elliptical, Parabolic, and Hyperbolic Orbits
As noted earlier, the relative orbit in the two-body problem is either an ellipse, parabola, or hyperbola depending on whether the energy is negative, zero, or positive, respectively. These curves are known collectively as conic sections and the generalization of Eq. (1) is
r = , (18)
where r and f have the same meaning as in Eq. (1), e is the generalized eccentricity, and p is a conserved quantity which depends upon the initial conditions. For an ellipse, p = a(1 [?] e [sup.2]), as in Eq. (1)). For a parabola, e = 1 and p = 2q, where q is the pericentric separation (distance of closest approach). For a hyperbola, e > 1 and p = q(1 + e), where q is again the pericentric separation. For all orbits, the three orientation angles i, O, and o are defined as in the elliptical case.
3. Planetary Perturbations and the Orbits of Small Bodies
Gravity is not restricted to interactions between the Sun and the planets or individual planets and their satellites, but rather all bodies feel the gravitational force of one another. Within the solar system, one body typically produces the dominant force on any given body, and the resultant motion can be thought of as a Keplerian orbit about a primary, subject to small perturbations by other bodies. In this section some important examples of the effects of these perturbations on the orbital motion are considered.
Classically, much of the discussion of the evolution of orbits in the solar system used perturbation theory as its foundation. Essentially, the method involves writing the equations of motion as the sum of a part that describes the independent Keplerian motion of the bodies about the Sun plus a part (called the disturbing function) that contains terms due to the pairwise interactions among the planets and minor bodies and the indirect terms associated with the back-reaction of the planets on the Sun. In general, one can then expand the disturbing function in terms of the small parameters of the problem (such as the ratio of the planetary masses to the solar mass, the eccentricities and inclinations, etc.), as well as the other orbital elements of the bodies, including the mean longitudes (i.e., the location of the bodies in their orbits), and attempt to solve the resulting equations for the time-dependence of the orbital elements.
3.1 Perturbed Keplerian Motion and Resonances
Although perturbations on a body's orbit are often small, they cannot always be ignored. They must be included in short-term calculations if high accuracy is required, for example, for predicting stellar occultations or targeting spacecraft. Most long-term perturbations are periodic in nature, their directions oscillating with the relative longitudes of the bodies or with some more complicated function of the bodies' orbital elements.
Small perturbations can produce large effects if the forcing frequency is commensurate or nearly commensurate with the natural frequency of oscillation of the responding elements. Under such circumstances, perturbations add coherently, and the effects of many small tugs can build up over time to create a large-amplitude, long-period response. This is an example of resonance forcing, which occurs in a wide range of physical systems.
An elementary example of resonance forcing is given by the simple one-dimensional harmonic oscillator, for which the equation of motion is
m + mG[sup.2] x = F [sub.o]cos pht. (19)
In Eq. (19), m is the mass of the oscillating particle, F [sub.o] is the amplitude of the driving force, G is the natural frequency of the oscillator, and ph is the forcing or resonance frequency. The solution to Eq. (19) is
x = x [sub.o]cos pht + Acos Gt + BsinGt, (20a)
x [sub.o] [?] , (20b)
and A and B are constants determined by the initial conditions. Note that if ph [?] G, a large-amplitude, long-period response can occur even if F [sub.o] is small. Moreover, if ph = G this solution to Eq. (19) is invalid. In this case the solution is given by
x = tsin Gt + Acos Gt + Bsin Gt. (21)
The t in front of the first term at the right-hand side of Eq. (21) leads to secular growth. Often this linear growth is moderated by the effects of nonlinear terms that are not included in the simple example provided here. However, some perturbations have a secular component.
Nearly exact orbital commensurabilities exist at many places in the solar system. Io orbits Jupiter twice as frequently as Europa does, which in turn orbits Jupiter twice as frequently as Ganymede does. Conjunctions (at which the bodies have the same longitude) always occur at the same position of Io's orbit (its perijove). How can such commensurabilities exist? After all, the probability of randomly picking a rational from the real number line is 0, and the number of small integer ratios is infinitely smaller still! The answer lies in the fact that orbital resonances may be held in place as stable locks, which result from nonlinear effects not represented in the foregoing simple mathematical example. For example, differential tidal recession (see Section 7.5) brings moons into resonance, and nonlinear interactions among the moons can keep them there.
Other examples of resonance locks include the Hilda asteroids, the Trojan asteroids, Neptune-Pluto, and the pairs of moons about Saturn, Mimas-Tethys and Enceladus-Dione. Resonant perturbation can also force material into highly eccentric orbits that may lead to collisions with other bodies; this is believed to be the dominant mechanism for clearing the Kirkwood gaps in the asteroid belt (see Section 5.1). Spiral density waves can result from resonant perturbations of a self-gravitating particle disk by an orbiting satellite. Density waves are seen at many resonances in Saturn's rings; they explain most of the structure seen in Saturn's A ring. The vertical analog of density waves, bending waves, are caused by resonant perturbations perpendicular to the ring plane due to a satellite in an orbit that is inclined to the ring. Spiral bending waves excited by the moons Mimas and Titan have been seen in Saturn's rings. In the next few subsections these manifestations of resonance effects that do not explicitly involve chaos are discussed. Chaotic motion produced by resonant forcing is discussed later in the chapter.
3.2 Examples of Resonances: Lagrangian Points, and Tadpole and Horseshoe Orbits
Many features of the orbits considered in this section can be understood by examining an idealized system in which two massive (but typically of unequal mass) bodies move on circular orbits about their common center of mass. If a third body is introduced that is much less massive than either of the first two, its motion can be followed by assuming that its gravitational force has no effect on the orbits of the other bodies. By considering the motion in a frame co-rotating with the massive pair (so that the pair remain fixed on a line that can be taken to be the x axis), Lagrange found that there are five points where particles placed at rest would feel no net force in the rotating frame. Three of the so-called Lagrange points (L [sub.1], L [sub.2], and L [sub.3]) lie along a line joining the two masses m [sub.1] and m [sub.2]. The other two Lagrange points (L [sub.4] and L [sub.5]) form equilateral triangles with the two massive bodies.
Particles displaced slightly from the first three Lagrangian points will continue to move away and hence these locations are unstable. The triangular Lagrangian points are potential energy maxima, which are stable for sufficiently large primary to secondary mass ratio due to the Coriolis force. Provided that the most massive body has at least 27 times the mass of the secondary (which is the case for all known examples in the solar system larger than the Pluto-Charon system), the Lagrangian points L [sub.4] and L [sub.5] are stable points. Thus, a particle at L [sub.4] or L [sub.5] that is perturbed slightly will start to "orbit" these points in the rotating coordinate system. Lagrangian points L [sub.4] and L [sub.5] are important in the solar system. For example, the Trojan asteroids in Jupiter's Lagrangian points and both Neptune and Mars confine their own Trojans. There are also small moons in the triangular Lagrangian points of Tethys and Dione, in the Saturnian system. The L [sub.4] and L [sub.5] points in the Earth-Moon system have been suggested as possible locations for space stations.
3.2.1 HORSESHOE AND TADPOLE ORBITS
Consider a moon on a circular orbit about a planet. Figure 3 shows some important dynamical features in the frame corotating with the moon. All five Lagrangian points are indicated in the picture. A particle just interior to the moon's orbit has a higher angular velocity than the moon in the stationary frame, and thus moves with respect to the moon in the direction of corotation. A particle just outside the moon's orbit has a smaller angular velocity, and moves away from the moon in the opposite direction. When the outer particle approaches the moon, the particle is slowed down (loses angular momentum) and, provided the initial difference in semimajor axis is not too large, the particle drops to an orbit lower than that of the moon. The particle then recedes in the forward direction. Similarly, the particle at the lower orbit is accelerated as it catches up with the moon, resulting in an outward motion toward the higher, slower orbit. Orbits like these encircle the L [sub.3], L [sub.4], and L [sub.5] points and are called horseshoe orbits. Saturn's small moons Janus and Epimetheus execute just such a dance, changing orbits every 4 years.
Since the Lagrangian points L [sub.4] and L [sub.5] are stable, material can librate about these points individually: such orbits are called tadpole orbits. The tadpole libration width at L [sub.4] and L [sub.5] is roughly equal to (m/M) [sup.1/2] r, and the horseshoe width is (m/M) [sup.1/3] r, where M is the mass of the planet, m the mass of the satellite, and r the distance between the two objects. For a planet of Saturn's mass, M = 5.7 x 10[sup.29] g, and a typical small moon of mass m = 10[sup.20] g (e.g., an object with a 30-km radius, with density of [?]1 g/cm[sup.3]), at a distance of 2.5 Saturnian radii, the tadpole libration half-width is about 3 km and the horseshoe half-width about 60 km.
FIGURE 3 Diagram showing the five Lagrangian equilibrium points (denoted by crosses) and three representative orbits near these points for the circular restricted three-body problem. In this example, the secondary's mass is 0.001 times the total mass. The coordinate frame has its origin at the barycenter and corotates with the pair of bodies, thereby keeping the primary (large solid circle) and secondary (small solid circle) fixed on the x axis. Tadpole orbits remain near one or the other of the L [sub.4] and L [sub.5] points. An example is shown near the L [sub.4] point on the diagram. Horseshoe orbits enclose all three of L [sub.3], L [sub.4], and L [sub.5] but do not reach L [sub.1] or L [sub.2]. The outermost orbit on the diagram illustrates this behavior. There is a critical curve dividing tadpole and horseshoe orbits that encloses L [sub.4] and L [sub.5] and passes through L [sub.3]. A horseshoe orbit near this dividing line is shown as the dashed curve in the diagram.
3.2.2 HILL SPHERE
The approximate limit to a planet's gravitational dominance is given by the extent of its Hill sphere,
R [sub.H] = a, (22)
where m is the mass of the planet and M is the Sun's mass. A test body located at the boundary of a planet's Hill sphere is subjected to a gravitational force from the planet comparable to the tidal difference between the force of the Sun on the planet and that on the test body. The Hill sphere essentially stretches out to the L [sub.1] point and is roughly the limit of the Roche lobe (maximum extent of an object held together by gravity alone) of a body with m [?] M. Planetocentric orbits that are stable over long periods of time are those well within the boundary of a planet's Hill sphere; all known natural satellites lie in this region. The trajectories
FIGURE 4 The orbit of J VIII Pasiphae, a distant retrograde satellite of Jupiter, is shown as seen in a nonrotating coordinate system with Jupiter at the origin (open circle). The satellite was integrated as a massless test particle in the context of the circular restricted three-body problem for approximately 38 years. The unit of distance is Jupiter's radius, R [sub.J]. During the course of this integration, the distance to Jupiter varied from 122 to 548R [sub.J]. Note how the large solar perturbations produce significant deviations from a Keplerian orbit. [Figure reprinted with permission from Jose Alvarellos (1996). "Orbital Stability of Distant Satellites of Jovian Planets," M.Sc. thesis, San Jose State University.]
of the outermost planetary satellites, which lie closest to the boundary of the Hill sphere, show large variations in planetocentric orbital paths (Fig. 4). Stable heliocentric orbits are those that are always well outside the Hill sphere of any planet.
3.3 Examples of Resonances: Ring Particles and Shepherding
In the discussions in Section 2, the gravitational force produced by a spherically symmetric body was described. In this section the effects of deviations from spherical symmetry must be included when computing the force. This is most conveniently done by introducing the gravitational potential Ph(r), which is defined such that the acceleration d [sup.2] r/dt [sup.2] of a particle in the gravitational field is
d [sup.2] r/dt [sup.2] = [?]Ph. (23)
In empty space, the Newtonian gravitational potential Ph(r) always satisfies Laplace's equation
[?][sup.2]Ph = 0. (24)
Most planets are very nearly axisymmetric, with the major departure from sphericity being due to a rotationally induced equatorial bulge. Thus, the gravitational potential can be expanded in terms of Legendre polynomials instead of the complete spherical harmonic expansion, which would be required for the potential of a body of arbitrary shape:
Ph(r, ph, th) = [?] [1 [?] J[sub.n]P[sub.n] (costh) (R/r)[sup.n] ]. (25)
This equation uses standard spherical coordinates, so that th is the angle between the planet's symmetry axis and the vector to the particle. The terms P[sub.n] (cos th) are the Legendre polynomials, and J[sub.n] are the gravitational moments determined by the planet's mass distribution. If the planet's mass distribution is symmetrical about the planet's equator, the J[sub.n] are zero for odd n. For large bodies, J [sub.2] is generally substantially larger than the other gravitational moments.
Consider a particle in Saturn's rings, which revolves around the planet on a circular orbit in the equatorial plane (th = 90deg) at a distance r from the center of the planet. The centripetal force must be provided by the radial component of the planet's gravitational force [see Eq. (9)], so the particle's angular velocity n satisfies
rn [sub.2](r) = . (26)
If the particle suffers an infinitesimal displacement from its circular orbit, it will oscillate freely in the horizontal and vertical directions about the reference circular orbit with radial (epicyclic) frequency k(r) and vertical frequency m(r), respectively, given by
k[sup.2] (r) = r [sup.[?]3][(r [sup.2] n)[sup.2]], (27)
m[sup.2](r) = . (28)
From Eqs. (24)-(28), the following relation is found between the three frequencies for a particle in the equatorial plane:
m[sup.2] = 2n [sup.2] [?] k[sup.2]. (29)
For a perfectly spherically symmetric planet, m = k = n. Since Saturn and the other ringed planets are oblate, m is slightly larger and k is slightly smaller than the orbital frequency n.
Using Eqs. (24)-(29), one can show that the orbital and epicyclic frequencies can be written as
n [sup.2] = [1 +3/2J [sub.2] [?] 15/8J [sub.4] + 35/16J [sub.6] + ...], (30)
k[sup.2] = [1 [?] 3/2J [sub.2] + 45/8J [sub.4] [?] 175/16 J [sub.6] + ...], (31)
m[sup.2] = [1+ 9/2J [sub.2] [?] 75/8 J [sub.4] + 245/16J [sub.6] + ...]. (32)
Thus, the oblateness of a planet causes apsides of particle orbits in and near the equatorial plane to precess in the direction of the orbit and lines of nodes of nearly equatorial orbits to regress.
Resonances occur where the radial (or vertical) frequency of the ring particles is equal to the frequency of a component of a satellite's horizontal (or vertical) forcing, as sensed in the rotating frame of the particle. In this case the resonating particle is always near the same phase in its radial (vertical) oscillation when it experiences a particular phase of the satellite's forcing. This situation enables continued coherent "kicks" from the satellite to build up the particle's radial (vertical) motion, and significant forced oscillations may thus result. The location and strengths of resonances with any given moon can be determined by decomposing the gravitational potential of the moon into its Fourier components. The disturbance frequency, ō, can be written as the sum of integer multiples of the satellite's angular, vertical, and radial frequencies:
p = jn [sub.s] + km[sub.s] + lk[sub.s], (33)
where the azimuthal symmetry number, j, is a nonnegative integer, and k and l are integers, with k being even for horizontal forcing and odd for vertical forcing. The subscript s refers to the satellite. A particle placed at distance r = r [sub.L] will undergo horizontal (Lindblad) resonance if r [sub.L] satisfies
p [?] jn(r [sub.L]) = +-m(r [sub.L]. (34)
It will undergo vertical resonance if its radial position r [sub.v], satisfies
p [?] jn(r [sub.L]) = +-m(r [sub.v]). (35)
When Eq. (34) is valid for the lower (upper) sign, r [sub.L] is referred to as the inner (outer) Lindblad or horizontal resonance. The distance r [sub.v] is called an inner (outer) vertical resonance if Eq. (35) is valid for the lower (upper) sign. Since all of Saturn's large satellites orbit the planet well outside the main ring system, the satellite's angular frequency n [sub.s] is less than the angular frequency of the particle, and inner resonances are more important than outer ones. When m [?] 1, the approximation m [?] n [?] k maybe used to obtain the ratio
= . (36)
The notation (j + k + l)/(j [?] 1) or (j + k + l):(j [?] 1) is commonly used to identify a given resonance.
The strength of the forcing by the satellite depends, to lowest order, on the satellite's eccentricity, e, and inclination, i, as e[sup.|l|] [sin i][sup.|k|] . The strongest horizontal resonances have k = l = 0, and are of the form j : (j [?] 1). The strongest vertical resonances have k = 1, l= 0, and are of the form (j + 1):(j [?] 1). The location and strengths of such orbital resonances can be calculated from known satellite masses and orbital parameters and Saturn's gravity field. Most strong resonances in the Saturnian system lie in the outer A ring near the orbits of the moons responsible for them. If n = m = k, the locations of the horizontal and vertical resonances would consider: r [sub.L] = r [sub.v]. Since, owing to Saturn's oblateness, m > n > k, the positions r [sub.L] and r [sub.v] do not coincide: r [sub.v] < r [sub.L]. A detailed discussion of spiral density waves, spiral bending waves, and gaps at resonances produced by moons is presented elsewhere in this encyclopedia. [See PLANETARY RINGS.]
4. Chaotic Motion
4.1 Concepts of Chaos
In the nineteenth century, Henri Poincare studied the mathematics of the circular restricted three-body problem. In this problem, one mass (the secondary) moves in a fixed, circular orbit about a central mass (the primary), while a test massless particle moves under the gravitational effect of both masses but does not perturb their orbits. From this work, Poincare realized that despite the simplicity of the equations of motion, some solutions to the problem exhibit complicated behavior.
Poincare's work in celestial mechanics provided the framework for the modern theory of nonlinear dynamics and ultimately led to a deeper understanding of the phenomenon of chaos, whereby dynamical systems described by simple equations can give rise to unpredictable behavior. The whole question of whether or not a given system is stable to sufficiently small perturbations is the basis of the Kolmogorov-Arnol'd-Moser (KAM) theory, which has its origins in the work of Poincare.
One characteristic of chaotic motion is that small changes in the starting conditions can produce vastly different final outcomes. Since all measurements of positions and velocities of objects in the solar system have finite accuracy, relatively small uncertainties in the initial state of the system can lead to large errors in the final state, for initial conditions that lie in chaotic regions in phase space.
This is an example of what has become known as the "butterfly effect," first mentioned in the context of chaotic weather systems. It has been suggested that under the right conditions, a small atmospheric disturbance (such as the flapping of a butterfly's wings) in one part of the world could ultimately lead to a hurricane in another part of the world.
The changes in an orbit that reveal it to be chaotic may occur very rapidly, for example during a close approach to the planet, or may take place over millions or even billions of years. Although there have been a number of significant mathematical advances in the study of nonlinear dynamics since Poincare's time, the digital computer has proven to be the most important tool in investigating chaotic motion in the solar system. This is particularly true in studies of the gravitational interaction of all the planets, where there are few analytical results.
4.2 The Three-body Problem as a Paradigm
The characteristics of chaotic motion are common to a wide variety of dynamical systems. In the context of the solar system, the general properties are best described by considering the planar circular restricted three-body problem, consisting of a massless test particle and two bodies of masses m [sub.1] and m [sub.2] moving in circular orbits about their common center of mass at constant separation with all bodies moving in the same plane. The test particle is attracted to each mass under the influence of the inverse square law of force given in Eq. (5). In Eq. (16), a is the constant separation of the two masses, and n = 2p/p is their constant angular velocity about the center of mass. Using x and y as components of the position vector of the test particle referred to the center of mass of the system (Fig. 5), the equations of motion of the particle in a reference frame
FIGURE 5 The rotating coordinate system used in the circular restricted three-body problem. The masses are at a fixed distance from one another and this is taken to be the unit of length. The position and velocity vectors of the test particle (at point P) are referred to the center of mass of the system at O.
rotating at angular velocity n are
x [?] 2ny [?] n [sup.2] x = [?]G (m [sub.1] [?] m [sub.2] ), (37)
y + 2nx [?] n [sup.2] y = [?]G y. (38)
where m[sub.1] = m [sub.1] a/(m [sub.1] + m [sub.2]), and m[sub.2] m [sub.2] a/(m [sub.1] + m [sub.2]) are constants and
r [sub.1] [sup.2] = (x + m[sub.2])[sup.2] + y [sup.2], (39)
r [sub.2] [sup.2] = (x [?] m[sub.1])[sup.2] + y [sup.2], (40)
where r [sub.1] and r [sub.2] are the distances of the test particle from the masses m [sub.1] and m [sub.2], respectively.
These two second-order, coupled, nonlinear differential equations can be solved numerically provided the initial position (x [sub.0], y [sub.0]) and velocity (x [sub.0], y [sub.0]) of the particle are known. Therefore the system is deterministic and at any given time the orbital elements of the particle (such as its semimajor axis and eccentricity) can be calculated from its initial position and velocity.
The test particle is constrained by the existence of a constant of the motion called the Jacobi constant, C, given by
C = n [sup.2](x [sup.2] + y [sup.2]) + 2G [?] x [sup.2] [?] y [sup.2]. (41)
The values of (x [sub.0], y [sub.0]) and (x [sub.0], y [sub.0]) fix the value of C for the system, and this value is preserved for all subsequent motion. At any instant the particle is at some position on the two-dimensional (x, y) plane. However, since the actual orbit is also determined by the components of the velocity (x, y), the particle can also be thought of as being at a particular position in a four-dimensional (x, y, x, y) phase space. Note that the use of four dimensions rather than the customary two is simply a means of representing the position and the velocity of the particle at a particular instant in time; the particle's motion is always restricted to the x [?] y plane. The existence of the Jacobi constant implies that the particle is not free to wander over the entire 4-D phase space, but rather that its motion is restricted to the 3-D "surface" defined by Eq. (41). This has an important consequence for studying the evolution of orbits in the problem.
The usual method is to solve the equations of motion, convert x, y, x, and y into orbital elements such as semi-major axis, eccentricity, longitude of periapse, and mean longitude, and then plot the variation of these quantities as a function of time. However, another method is to produce a surface of section, also called a Poincare map. This makes use of the fact that the orbit is always subject to Eq. (41), where C is determined by the initial position and velocity. Therefore if any three of the four quantities x, y, x, and y are known, the fourth can always be determined by solving Eq. (41). One common surface of section that can be obtained for the planar circular restricted three-body problem is a plot of values of x and x whenever y = 0 and y is positive. The actual value of y can always be determined uniquely from Eq. (41), and so the two-dimensional (x, x) plot implicitly contains all the information about the particle's location in the four-dimensional phase space. Although surfaces of section make it more difficult to study the evolution of the orbital elements, they have the advantage of revealing the characteristic motion of the particle (regular or chaotic) and a number of orbits can be displayed on the same diagram.
As an illustration of the different types of orbits that can arise, the results of integrating a number of orbits using a mass m [sub.2]/(m [sub.1] + m [sub.2]) = 10[sup.[?]3] and Jacobi constant C = 3.07 are described next. In each case, the particle was started with the initial longitude of periapse p[sub.0] = 0 and initial mean longitude l[sub.0] = 0. This corresponds to x = 0 and y = 0. Since the chosen mass ratio is comparable to that of the Sun-Jupiter system, and Jupiter's eccentricity is small, this
FIGURE 6 The eccentricity as a function of time for an object moving in a regular orbit near the 7:4 resonance with Jupiter. The plot was obtained by solving the circular restricted three-body problem numerically using initial values of 0.6944 and 0.2065 for the semimajor axis and eccentricity, respectively. The corresponding position and velocity in the rotating frame were x [sub.0] = 0.55, y [sub.0] = 0, x = 0, and y = 0.9290.
will be used as a good approximation to the motion of fictitious asteroids moving around the Sun under the effect of gravitational perturbations from Jupiter. The asteroid is assumed to be moving in the same plane as Jupiter's orbit.
4.2.1 REGULAR ORBITS
The first asteroid has starting values x = 0.55, y = 0, x = 0, with y = 0.9290 determined from the solution of Eq. (41). Here a set of dimensionless coordinates are used in which n = 1, G = 1, and m [sub.1] + m [sub.2] = 1. In these units, the orbit of m2 is a circle at distance a = 1 with uniform speed v = 1. The corresponding initial values of the heliocentric semimajor axis and eccentricity are a [sub.0] =0.6944 and e [sub.0] =0.2065. Since the semimajor axis of Jupiter's orbit is 5.202 AU, this value of a [sub.0] would correspond to an asteroid at 3.612 AU.
Figure 6 shows the evolution of e as a function of time. The plot shows a regular behavior with the eccentricity varying from 0.206 to 0.248 over the course of the integration. In fact, an asteroid at this location would be close to an orbit-orbit resonance with Jupiter, where the ratio of the orbital period of the asteroid, T, to Jupiter's period, T [sub.J], is close to a rational number. From Kepler's third law of planetary motion, T [sup.2] [?] a [sup.3]. In this case, T/T [sub.J] = (a/a [sub.J])[sup.3/2] = 0.564 [?] 4/7 and the asteroid orbit is close to a 7:4 resonance with Jupiter. Figure 7 shows the variation of the semimajor axis of the asteroid, a, over the same time interval as shown in Fig. 6. Although the changes in a are correlated with those in e, they are smaller in amplitude and a appears to oscillate about the location of the exact resonance at a = (4/7)[sup.2/3] [?] 0.689. An asteroid in resonance experiences enhanced gravitational perturbations from Jupiter, which can cause regular variations in its orbital elements. The extent of these variations depends on the asteroid's location within the resonance, which is, in turn, is determined by the starting conditions.
FIGURE 7 The semimajor axis as a function of time for an object using the same starting conditions as in Fig. 6. The units of the semimajor axis are such that Jupiter's semimajor axis (5.202 AU) is taken to be unity.
The equations of motion can be integrated with the same starting conditions to generate a surface of section by plotting the values of x and x whenever y = 0 with y > 0 (Fig. 8). The pattern of three distorted curves or "islands" that emerges is a characteristic of resonant motion when displayed in such plots. If a resonance is of the form (p + q):p, where p and q are integers, then q is said to be the order of the resonance. The number of islands seen in a surface of section plot of a given resonant trajectory is equal to q. In this case, p = 4, q = 3 and three islands are visible.
The center of each island would correspond to a starting condition that placed the asteroid at exact resonance where the variation in e and a would be minimal. Such points are said to be fixed points of the Poincare map. If the starting location was moved farther away from the center, the subsequent variations in e and a would get larger, until eventually some starting values would lead to trajectories that were not in resonant motion.
FIGURE 8 A surface of section plot for the same (regular) orbit shown in Figs. 6 and 7. The 2000 points were generated by plotting the values of x and x whenever y = 0 with positive y. The three "islands" in the plot are due to the third-order 7:4 resonance.
4.2.2 CHAOTIC ORBITS
Figures 9 and 10 show the plots of e and a as a function of time for an asteroid orbit with starting values x [sub.0] = 0.56, y [sub.0] = 0, x [sub.0] = 0, and y determined from Eq. (41) with C = 3.07. The corresponding orbital elements are a [sub.0] = 0.6984 and e [sub.0] = 0.1967. These values are only slightly different from those used earlier, indeed the initial behavior of the plots is quite similar to that seen in Figs. 6 and 7. However, subsequent variations in e and a are strikingly different. The eccentricity varies from 0.188 to 0.328 in an irregular manner, and the value of a is not always close to the value associated with exact resonance. This is an example of a chaotic trajectory where the variations in the orbital elements have no obvious periodic or quasi-periodic structure. The anticorrelation of a and e can be explained in terms of the Jacobi constant.
The identification of this orbit as chaotic becomes apparent from a study of its surface of section (Fig. 11). Clearly, this orbit covers a much larger region of phase space than the previous example. Furthermore, the orbit does not lie on a smooth curve, but is beginning to fill an area of the phase space. The points also help to define a number of empty regions, three of which are clearly associated with the 7:4 resonance seen in the regular trajectory. There is also a tendency for the points to "stick" near the edges of the islands; this gives the impression of regular motion for short periods of time.
Chaotic orbits have the additional characteristic that they are sensitively dependent on initial conditions. This is illustrated in Fig. 12, where the variation in e as a function of time is shown for two trajectories; the first corresponds to Fig. 9 (where x [sub.0] = 0.56) and the second has x [sub.0] = 0.56001. The initial value of y was chosen so that the same value of C was obtained. Although both trajectories show comparable initial variations in e, after 60 Jupiter periods it is clear that the orbits have drifted apart. Such a divergence would not occur for nearby orbits in a regular part of the phase space.
FIGURE 9 The eccentricity as a function of time for an object moving in a chaotic orbit started just outside the 7:4 resonance with Jupiter. The plot was obtained by solving the circular restricted three-body problem numerically using initial values of 0.6984 and 0.1967 for the semimajor axis and eccentricity, respectively. The corresponding position and velocity in the rotating frame were x [sub.0] = 0.56, y [sub.0] = 0, x [sub.0] = 0, and y = 0.8998.
The rate of divergence of nearby trajectories in such numerical experiments can be quantified by monitoring the evolution of two orbits that are started close together. In a dynamical system such as the three-body problem, there are a number of quantities called the Lyapunov characteristic exponents. A measurement of the local divergence of nearby trajectories leads to an estimate of the largest of these exponents, and this can be used to determine whether or not the system is chaotic. If two orbits are separated in phase space by a distance d [sub.0] at time t [sub.0], and d is their separation at time t, then the orbit is chaotic if
d = d [sub.0] exp g (t [?] t [sub.0]), (42)
where g is a positive quantity equal to the maximum Lyapunov characteristic exponent. However, in practice the Lyapunov characteristic exponents can only be derived analytically for a few idealized systems. For practical problems in the solar system, g can be estimated from the results
FIGURE 10 The semimajor axis as a function of time for an object using the same starting conditions as in Fig. 9. The units of the semimajor axis are such that Jupiter's semimajor axis (5.202 AU) is taken to be unity.
of a numerical integration by writing
g = (43)
and monitoring the behavior of g with time. A plot of g as a function of time on a log-log scale reveals a striking difference between regular and chaotic trajectories. For regular orbits, d [?] d [sup.0] and a log-log plot has a slope of [?]1. However, if the orbit is chaotic, then g tends to a constant non-zero value. This method may not always work because g is defined only in the limit as t - [?] and sometimes chaotic orbits may give the appearance of being regular orbits for long periods of time by sticking close to the edges of the islands.
If the nearby trajectory drifts too far from the original one, then g is no longer a measure of the local divergence of the orbits. To overcome this problem, it helps to rescale the separation of the nearby trajectory at fixed intervals. Figure 13 shows log as a function of log t calculated using this
FIGURE 11 A surface of section plot for the same chaotic orbit as shown in Figs. 9 and 10. The 2000 points were generated by plotting the values of x and x whenever y = 0 with positive y. The points are distributed over a much wider region of the (x, x) plane than the points for the regular orbit shown in Fig. 8, and they help to define the edges of the regular regions associated with the 7:4 and other resonances.
FIGURE 12 The variation in the eccentricity for two chaotic orbits started close to one another. One plot is part of Fig. 9 using the chaotic orbit started with x [sub.0] = 0.56, and the other is for an orbit with x [sub.0] = 0.56001. Although the divergence of the two orbits is exponential, the effect becomes noticeable only after 60 Jupiter periods.
FIGURE 13 The evolution of the quantity g [defined in Eq. (43)] as a function of time (in Jupiter periods) for a regular (x [sub.0] = 0.55) and chaotic (x [sub.0] = 0.56) orbit. In this log-log plot, the regular orbit shows a characteristic slope of [?]1 with no indication of log g tending toward a finite value. However, in the case of the chaotic orbit, log g tends to a limiting value close to [?]0.77.
method for the regular and chaotic orbits described here. This leads to an estimate of g = 10[sup.[?]0.77] (Jupiter periods)[sup.[?]1] for the maximum Lyapunov characteristic exponent of the chaotic orbit. The corresponding Lyapunov time is given by 1/g, or in this case [?]6 Jupiter periods. This indicates that for this starting condition the chaotic nature of the orbit quickly becomes apparent.
It is important to realize that a chaotic orbit is not necessarily unbounded. The maximum Lyapunov characteristic exponent concerns local divergence and provides no information about the global stability of the trajectory. The phrase "wandering on a leash" is an apt description of objects on bounded chaotic orbits--the motion is contained but yet chaotic at the same time. Another consideration is that numerical explorations of chaotic systems have many pitfalls both in how the physical system is modeled and whether or not the model provides an accurate portrayal of the real system.
4.2.3 LOCATION OF REGULAR AND CHAOTIC REGIONS
The extent of the chaotic regions of the phase space of a dynamical system can depend on a number of factors. In the case of the circular restricted three-body problem, the critical quantities are the values of the Jacobi constant and the mass ratio m[sub.2]. In Figs. 14 and 15, ten trajectories are shown for each of two different values of the Jacobi constant. In the first case (Fig. 14), the value is C = 3.07 (the same as the value used in Figs. 8 and 11), whereas in Fig. 15 it is C = 3.13. It is clear that the extent of the chaos is reduced in Fig. 15. The value of C in the circular restricted problem determines how close the asteroid can get to Jupiter. Larger values of C correspond to orbits with greater minimum dis-
FIGURE 14 Representative surface of section plots for x [sub.0] = 0.25, 0.29, 0.3, 0.45, 0.475, 0.5, 0.55, 0.56, 0.6, and 0.8 with = 0, y [sub.0] = 0, and Jacobi constant C = 3.07. Each trajectory was followed for a minimum of 500 crossing points. The plot uses the points shown in Figs. 8 and 11 (although the scales are different), as well as points from other regular and chaotic orbits. The major resonances are identified.
tances from Jupiter. For the case m[sub.2] = 0.001 and C > 3.04, it is impossible for their orbits to intersect, although the perturbations can still be significant.
Close inspection of the separatrices in Figs. 14 and 15 reveals that they consist of chaotic regions with regular regions on either side. As the value of the Jacobi constant decreases, the extent of the chaotic separatrices increases until the regular curves separating adjacent resonances are broken down and neighboring chaotic regions begin to merge. This can be thought of as the overlap of adjacent resonances giving rise to chaotic motion. It is this process that permits chaotic orbits to explore regions of the phase space that are inaccessible to the regular orbits. In the context of the Sun-Jupiter-asteroid problem, this observation implies that asteroids in certain orbits are capable of large excursions in their orbital elements.
5. Orbital Evolution of Minor Bodies
With more than 130,000 accurately determined orbits and one major perturber (the planet Jupiter), the asteroids provide a natural laboratory in which to study the consequences of regular and chaotic motion. Using suitable approximations, asteroid motion can be studied analytically in some special cases. However, it is frequently necessary to resort to numerical integration. [See MAIN-BELT ASTEROIDS.]
Investigations have shown that a number of asteroids have orbits that result in close approaches to planets. Of particular interest are asteroids such as 433 Eros, 1033 Ganymed, and 4179 Toutatis, because they are on orbits
FIGURE 15 Representative surface of section plots for x [sub.0] = 0.262, 0.3, 0.34, 0.35, 0.38, 0.42, 0.52, 0.54, 0.7, and 0.78 with x [sub.0] = 0, y[sub.0] = 0, and Jacobi constant C = 3.13. Each trajectory was followed for a minimum of 500 crossing points. It is clear from a comparison with Fig. 14 that the phase space is more regular; chaotic orbits still exist for this value of C, but they are more difficult to find. The major resonances are identified.
that bring them close to Earth. One of the most striking examples of the butterfly effect (see Section 4.1) in the context of orbital evolution is the orbit of asteroid 2060 Chiron, which has a perihelion inside Saturn's orbit and an aphelion close to Uranus's orbit. Numerical integrations based on the best available orbital elements show that it is impossible to determine Chiron's past or future orbit with any degree of certainty since it frequently suffers close approaches to Saturn and Uranus. In such circumstances, the outcome is strongly dependent on the initial conditions as well as the accuracy of the numerical method. These are the characteristic signs of a chaotic orbit. By integrating several orbits with initial conditions close to the nominal values, it is possible to carry out a statistical analysis of the orbital evolution. Studies suggest that there is a 1 in 8 chance that Saturn will eject Chiron from the solar system on a hyperbolic orbit, while there is a 7 in 8 chance that it will evolve toward the inner solar system and come under strong perturbations from Jupiter. Telescopic observations of a faint coma surrounding Chiron imply that it is a comet rather than an asteroid; perhaps its future orbit will resemble that of a short-period comet of the Jupiter family.
Numerical studies of the orbital evolution of planet-crossing asteroids under the effects of perturbations from all the planets have shown a remarkable complexity of motion for some objects. For example, the Earth-crossing asteroid 1620 Geographos gets trapped temporarily in a number of resonances with Earth in the course of its chaotic evolution (Fig. 16).
A histogram of the number distribution of asteroid orbits in semimajor axis (Fig. 17) shows that apart from a clustering of asteroids near Jupiter's semimajor axis at 5.2 AU, there is an absence of objects within 0.75 AU of the orbit of Jupiter. The objects in the orbit of Jupiter are the Trojan asteroids (Section 3.2), which are located [?]60deg ahead of and behind Jupiter.
The cleared region near Jupiter's orbit can be understood in terms of chaotic motion due to the overlap of adjacent resonances. In the context of the Sun-Jupiter-asteroid restricted three-body problem, the perturber (Jupiter) has
FIGURE 16 A plot of the semimajor axis of the near-Earth asteroid 1620 Geographos over a backward and forward integration of 100,000 years starting in 1986. Under perturbations from the planets, Geographos moves in a chaotic orbit and gets temporarily trapped in a number of high-order, orbit-orbit resonances (indicated in the diagram) with Earth. The data are taken from a numerical study of planet-crossing asteroids undertaken by A. Milani and coworkers. (Courtesy of Academic Press.)
FIGURE 17 A histogram of the distribution of the numbered asteroids with semimajor axis together with the locations of the major jovian resonances. Most objects lie in the main belt between 2.0 and 3.3 AU, where the outer edge is defined by the location of the 2:1 resonance with Jupiter. As well as gaps (the Kirkwood gaps) at the 3:1, 5:2, 2:1, and other resonances in the main belt, there are small concentrations of asteroids at the 3:2 and 1:1 resonances (the Hilda and Trojan groups, respectively).
an infinite sequence of first-order resonances that lie closer together as its semimajor axis is approached. For example, the 2:1, 3:2, 4:3, and 5:4 resonances with Jupiter lie at 3.3, 4.0, 4.3, and 4.5 AU, respectively. Since each (p + 1):p resonance (where p is a positive integer) has a finite width in semimajor axis that is almost independent of p, adjacent resonances will always overlap for some value of p greater than a critical value, p [sub.crit]. This value is given by
p [sub.crit] [?] 0.51 (44)
FIGURE 18 The chaotic evolution of the eccentricity of a fictitious asteroid at the 3:1 resonance with Jupiter. The orbit was integrated using an algebraic mapping technique developed by J. Wisdom. The line close to e = 0.3 denotes the value of the asteroid's eccentricity, above which it would cross the orbit of Mars. It is believed that the 3:1 Kirkwood gap was created when asteroids in chaotic zones at the 3:1 resonance reached high eccentricities and were removed by direct encounters with Mars, Earth, or Venus.
where, in this case, m is the mass of Jupiter and M is the mass of the Sun. This equation can be used to predict that resonance overlap and chaotic motion should occur for p values greater than 4; this corresponds to a semimajor axis near 4.5 AU. Therefore chaos may have played a significant role in the depletion of the outer asteroid belt.
The histogram in Fig. 17 also shows a number of regions in the main belt where there are few asteroids. The gaps at 2.5 and 3.3 AU were first detected in 1867 by Daniel Kirkwood using a total sample of fewer than 100 asteroids; these are now known as the Kirkwood gaps. Their locations coincide with prominent Jovian resonances (indicated in Fig. 17), and this led to the hypothesis that they were created by the gravitational effect of Jupiter on asteroids that had orbited at these semimajor axes. The exact removal mechanism was unclear until the 1980s, when several numerical and analytical studies showed that the central regions of these resonances contained large chaotic zones.
The Kirkwood gaps cannot be understood using the model of the circular restricted three-body problem described in Section 4.2. The eccentricity of Jupiter's orbit, although small (0.048), plays a crucial role in producing the large chaotic zones that help to determine the orbital evolution of asteroids. On timescales of several hundreds of thousands of years, the mutual perturbations of the planets act to change their orbital elements and Jupiter's eccentricity can vary from 0.025 to 0.061. An asteroid in the chaotic zone at the 3:1 resonance would undergo large, essentially unpredictable changes in its orbital elements. In particular, the eccentricity of the asteroid could become large enough for it to cross the orbit of Mars. This is illustrated in Fig. 18 for a fictitious asteroid with an initial eccentricity of 0.15 moving in a chaotic region of the phase space at the 3:1 resonance. Although the asteroid can have periods of relatively low eccentricity, there are large deviations and
FIGURE 19 The effect of an increase in the orbital eccentricity of an asteroid at the 3:1 Jovian resonance on the closest approach between the asteroid and Mars. For e = 0.15, the orbits do not cross. However, for e = 0.33, a typical maximum value for asteroids in chaotic orbits, there is a clear intersection of the orbits, and the asteroid could have a close encounter with Mars.
e can reach values in excess of 0.3. Allowing for the fact that the eccentricity of Mars's orbit can reach 0.14, this implies that there will be times when the orbits could intersect (Fig. 19). In this case, the asteroid orbit would be unstable, since it is likely to either impact the surface of Mars or suffer a close approach that would drastically alter its semimajor axis. Although Jupiter provides the perturbations, it is Mars, Earth or Venus that ultimately removes the asteroids from the 3:1 resonance. Figure 20 shows the excellent correspondence between the distribution of asteroids close to the 3:1 resonance and the maximum extent of the chaotic region determined from numerical experiments.
The situation is less clear for other resonances, although there is good evidence for large chaotic zones at the 2:1 and 5:2 resonances. In the outer part of the main belt, large changes in eccentricity will cause the asteroid to cross the orbit of Jupiter before it gets close to Mars. There may also
FIGURE 20 The eccentricity and semimajor axes of asteroids in the vicinity of the 3:1 jovian resonance; the Kirkwood gap is centered close to 2.5 AU. The two curves denote the maximum extent of the chaotic zone determined from numerical experiments, and there is excellent agreement between these lines and the edges of the 3:1 gap.
be perturbing effects from other planets. In fact, it is now known that secular resonances have an important role to play in the clearing of the Kirkwood gaps, including the one at the 3:1 resonance. Once again, chaos is involved. Studies of asteroid motion at the 3:2 Jovian resonance indicate that the motion is regular, at least for low values of the eccentricity. This may help to explain why there is a local concentration of asteroids (the Hilda group) at this resonance, whereas others are associated with an absence of material.
Since the dynamical structure of the asteroid belt has been determined by the perturbative effects of nearby planets, it seems likely that the original population was much larger and more widely dispersed. Therefore, the current distribution of asteroids may represent objects that are either recent collision products or that have survived in relatively stable orbits over the age of the solar system.
Most meteorites are thought to be the fragments of material produced from collisions in the asteroid belt, and the reflectance properties of certain meteorites are known to be similar to those of common types of asteroids. Since most collisions take place in the asteroid belt, the fragments have to evolve into Earth-crossing orbits before they can hit Earth and be collected as samples.
An estimate of the time taken for a given meteorite to reach Earth after the collisional event that produced it can be obtained from a measure of its cosmic ray exposure age. Prior to the collisions, the fragment may have been well below the surface of a much larger body, and as such it would have been shielded from all but the most energetic cosmic rays. However, after a collision the exposed fragment would be subjected to cosmic ray bombardment in interplanetary space. A detailed analysis of meteorite samples allows these exposure ages to be measured.
In the case of one common class of meteorites called the ordinary chondrites, the cosmic ray exposure ages are typically less than 20 million years and the samples show little evidence of having been exposed to high pressure, or "shocking." Prior to the application of chaos theory to the origin of the Kirkwood gaps, there was no plausible mechanism that could explain delivery to Earth within the exposure age constraints and without shocking. However, small increments in the velocity of the fragments as a result of the initial collision could easily cause them to enter a chaotic zone near a given resonance. Numerical integrations of such orbits near the 3:1 resonance showed that it was possible for them to achieve eccentricities large enough for them to cross the orbit of Earth. This result complemented previous research that had established that this part of the asteroid belt was a source region for the ordinary chondrites. Another effect that must be considered to obtain agreement between theory and observations is the Yarkovski effect which is discussed below. [See METEORITES.]
Typical cometary orbits have large eccentricities and therefore planet-crossing trajectories are commonplace. Many comets are thought to originate in the Oort cloud at several tens of thousands of AU from the Sun; another reservoir of comets, known as the Kuiper belt, exists just beyond the orbit of Neptune. Those that have been detected from Earth are classified as either long period (most of which have made single apparitions and have periods >200 yr) or Halley-type (with orbital periods of 20-200 yr) or Jupiter-family, which have orbital periods <20 yr. All comets with orbital periods of less than [?]10[sup.3] yr have experienced a close approach to Jupiter or one of the other giant planets. By their very nature, the orbits of comets are chaotic, since the outcome of any planetary encounter will be sensitively dependent on the initial conditions.
Studies of the orbital evolution of the short-period comet P/Lexell highlight the possible effects of close approaches. A numerical integration has shown that prior to 1767 it was a short-period comet with a semimajor axis of 4.4 AU and an eccentricity of 0.35. In 1767 and 1779, it suffered close approaches to Jupiter. The first encounter placed it on a trajectory which brought it into the inner solar system and close (0.0146 AU) to the Earth, leading to its discovery and its only apparition in 1770, whereas the second was at a distance of [?]3 Jovian radii. This changed its semimajor axis to 45 AU with an eccentricity of 0.88.
A more recent example is the orbital history of comet Shoemaker-Levy 9 prior to its spectacular collision with Jupiter in 1994. Orbit computations suggest that the comet was first captured by Jupiter at some time during a 9-year interval centered on 1929. Prior to its capture, it is likely that it was orbiting in the outer part of the asteroid belt close to the 3:2 resonance with Jupiter or between Jupiter and Saturn close to the 2:3 resonance with Jupiter. However, the chaotic nature of its orbit means that it is impossible to derive a more accurate history unless prediscovery images of the comet are obtained. [See PHYSICS AND CHEMISTRY OF COMETS; COMETARY DYNAMICS.]
5.4 Small Satellites and Rings
Chaos is also involved in the dynamics of a satellite embedded in a planetary ring system. The processes differ from those discussed in Section 3.1, A because there is a near-continuous supply of ring material and direct scattering by the perturber is now important. In this case, the key quantity is the Hill's sphere of the satellite. Ring particles on near-circular orbits passing close to the satellite exhibit chaotic behavior due to the significant perturbations they receive at close approach. This causes them to collide with surrounding ring material, thereby forming a gap. Studies have shown that for small satellites, the expression for the width of the cleared gap is
W [?] 0.44 a (45)
where m [sub.2] and a are the mass and semimajor axis of the satellite and m [sub.1] is the mass of the planet. Thus, an icy satellite with a radius of 10 km and a density of 1 g cm[sup.[?]3] orbiting in Saturn's A ring at a radial distance of 135,000 km would create a gap approximately 140 km wide.
Since such a gap is wider than the satellite that creates it, this provides an indirect method for the detection of small satellites in ring systems. There are two prominent gaps in Saturn's A ring: the [?]35-km-wide Keeler gap at 135,800 km and the 320-km-wide Encke gap at 133,600 km. The predicted radii of the icy satellites required to produce these gaps are [?]2.5 and [?]24 km, respectively. In 1991, an analysis of Voyager images by M. Showalter revealed a small satellite, Pan, with a radius of [?]10 km orbiting in the Encke gap. In 2005, the moon Daphnis of radius [?]3-4 km was discovered in the Keeler gap by the Cassini spacecraft. Voyager 2 images of the dust rings of Uranus show pronounced gaps at certain locations. Although most of the proposed shepherding satellites needed to maintain the narrow rings have yet to be discovered, these gaps may provide indirect evidence of their orbital locations.
6. Long Term Stability of Planetary Orbits
6.1 The N-Body Problem
The entire solar system can be approximated by a system of nine planets orbiting the Sun. (Tiny Pluto has been included in most studies of this problem to date, because it was classified as a planet until 2006. But Pluto does not substantially perturb the motions of the eight larger planets.) In a center of mass frame, the vector equation of motion for planet i moving under the Newtonian gravitational effect of the Sun and the remaining 8 planets is given by
r̈ = G m[sub.j] (j [?] i), (46)
where r [sub.i ] and m[sub.i] are the position vector and mass of planet i(i = 1, 2,..., 9), respectively, r [sub.ij ] [?] r [sub.j ] [?] r [sub.i ], and the subscript 0 refers to the Sun. These are the equations of the N-body problem for the case where N = 10, and although they have a surprisingly simple form, they have no general, analytical solution. However, as in the case of the three-body problem, it is possible to tackle this problem mathematically by making some simplifying assumptions.
Provided the eccentricities and inclinations of the N bodies are small and there are no resonant interactions between the planets, it is possible to derive an analytical solution that describes the evolution of all the eccentricities, inclinations, perihelia, and nodes of the planets. This solution, called Laplace-Lagrange secular perturbation theory, gives no positional information about the planets, yet it demonstrates that there are long-period variations in the planetary orbital elements that arise from mutual perturbations. The secular periods involved are typically tens or hundreds of thousands of years, and the evolving system always exhibits a regular behavior. In the case of Earth's orbit, such periods may be correlated with climatic change, and large variations in the eccentricity of Mars are thought to have had important consequences for its climate.
In the early nineteenth century, Pierre Simon de Laplace claimed that he had demonstrated the long-term stability of the solar system using the results of his secular perturbation theory. Although the actual planetary system violates some of the assumed conditions (e.g., Jupiter and Saturn are close to a 5:2 resonance), the Laplace-Lagrange theory can be modified to account for some of these effects. However, such analytical approaches always involve the neglect of potentially important interactions between planets. The problem becomes even more difficult when the possibility of near-resonances between some of the secular periods of the system is considered. However, nowadays it is always possible to carry out numerical investigations of long-term stability.
6.2 Stability of the Solar System
Numerical integrations show that the orbits of the planets are chaotic, although there is no indication of gross instability in their motion provided that the integrations are restricted to durations of 5 billion years (the age of the solar system). The eight planets as well as dwarf planet Pluto remain more or less in their current orbits with small, nearly periodic variations in their eccentricities and inclinations; close approaches never seem to occur. Pluto's orbit is chaotic, partly as a result of its 3:2 resonance with the planet Neptune, although the perturbing effects of other planets are also important. Despite the fact that the timescale for exponential divergence of nearby trajectories (the inverse of the Lyapunov exponent) is about 20 million years, no study has shown evidence for Pluto leaving the resonance.
Chaos has also been observed in the motion of the eight planets, and it appears that the solar system as a whole is chaotic with a timescale for exponential divergence of 4 or 5 million years, although different integrations give different results. However, the effect is most apparent in the orbits of the inner planets. Though there appear to be no dramatic consequences of this chaos, it does mean that the use of the deterministic equations of celestial mechanics to predict the future positions of the planets will always be limited by the accuracy with which their orbits can be measured. For example, some results suggest that if the position of Earth along its orbit is uncertain by 1 cm today, then the exponential propagation of errors that is characteristic of chaotic motion implies that knowledge of Earth's orbital position 200 million years in the future is not possible.
The solar system appears to be "stable" in the sense that all numerical integrations show that the planets remain close to their current orbits for timescales of billions of years. Therefore the planetary system appears to be another example of bounded chaos, where the motion is chaotic but always takes place within certain limits. Although an analytical proof of this numerical result and a detailed understanding of how the chaos has arisen have yet to be achieved, the solar system seems to be chaotic yet stable. When the planetary orbits are integrated forward for timescales for several billion years using the averaged equations of motion, it is found that there is a very small but finite probability that the orbit of Mercury can become unstable and intersect the orbit of Venus. Many challenges remain in understanding how structural stability of planetary systems in the presence of transient and intermittent chaos can be maintained, and this subject remains a rich field for dynamical exploration.
7. Dissipative Forces and the Orbits of Small Bodies
The foregoing sections describe the gravitational interactions between the Sun, planets, and moons. Solar radiation has been ignored, but this is an important force for small particles in the solar system. Three effects can be distinguished: (1) the radiation pressure, which pushes particles primarily outward from the Sun (micron-sized dust); (2) the Poynting-Robertson drag, which causes centimeter-sized particles to spiral inward toward the Sun; and (3) the Yarkovski effect, which changes the orbits of meter- to kilometer-sized objects owing to uneven temperature distributions at their surfaces. The latter two effects are relativistic and thus quite weak at solar system velocities, but they can nonetheless be significant as they can lead to secular changes in orbital angular momentum and energy. Each of these effects is discussed in the next three subsections and then the effect of gas drag is examined. In the final subsection the influence of tidal interactions is discussed; this effect (in contrast to the other dissipative effects described in this section) is most important for larger bodies such as moons and planets. [See SOLAR SYSTEM DUST.]
7.1 Radiation Force (Micron-Sized Particles)
The Sun's radiation exerts a force, F [sub.r], on all other bodies of the solar system. The magnitude of this force is
F [sub.r] = Q [sub.pr], (47)
where A is the particle's geometric cross section, L is the solar luminosity, c is the speed of light, r is the heliocentric distance, and Q [sub.pr] is the radiation pressure coefficient, which is equal to unity for a perfectly absorbing particle and is of order unity unless the particle is small compared to the wavelength of the radiation. The parameter b is defined as the ratio between the forces due to the radiation pressure and the Sun's gravity:
b [?] = 5.7 x 10[sup.[?]5] , (48)
where the radius, R, and the density, r, of the particle are in c.g.s. units. Note that b is independent of heliocentric distance and that the solar radiation force is important only for micron- and submicron-sized particles. Using the parameter b, a more general expression for the effective gravitational attraction can be written:
F [sub.geff] = , (49)
that is, the small particles "see" a Sun of mass (1 [?] b)M. It is clear that small particles with b > 1 are in sum repelled by the Sun, and thus quickly escape the solar system, unless they are gravitationally bound to one of the planets. Dust which is released from bodies traveling on circular orbits at the Keplerian velocity is ejected from the solar system if b > 0.5.
The importance of solar radiation pressure can be seen, for example, in comets. Cometary dust is pushed in the antisolar direction by the Sun's radiation pressure. The dust tails are curved because the particles' velocity decreases as they move farther from the Sun, due to conservation of angular momentum. [See COMETARY DYNAMICS; PHYSICS AND CHEMISTRY OF COMETS.]
7.2 Poynting-Robertson Drag (Centimeter-Sized Grains)
A small particle in orbit around the Sun absorbs solar radiation and reradiates the energy isotropically in its own frame. The particle thereby preferentially radiates (and loses momentum) in the forward direction in the inertial frame of the Sun. This leads to a decrease in the particle's energy and angular momentum and causes dust in bound orbits to spiral sunward. This effect is called the Poynting-Robertson drag.
The net force on a rapidly rotating dust grain is given by
F [sub.rad] [?] [(1 [?] )r̂ [?] tĥ]. (50)
The first term in Eq. (50) is that due to radiation pressure and the second and third terms (those involving the velocity of the particle) represent the Poynting-Robertson drag.
From this discussion, it is clear that small-sized dust grains in the interplanetary medium are removed: (sub)-micron sized grains are blown out of the solar system, whereas larger particles spiral inward toward the Sun. Typical decay times (in years) for circular orbits are given by
t[sub.P[?]R] [?] 400 , (51)
with the distance r in AU.
Particles that produce the bulk of the zodiacal light (at infrared and visible wavelengths) are between 20 and 200 mm, so their lifetimes at Earth orbit are on the order of 10[sup.5] yr, which is much less than the age of the solar system. Sources for the dust grains are comets as well as the asteroid belt, where numerous collisions occur between countless small asteroids.
7.3 Yarkovski Effect (Meter-Sized Objects)
Consider a rotating body heated by the Sun. Because of thermal inertia, the afternoon hemisphere is typically warmer than the morning hemisphere, by an amount DT [?] T. Let us assume that the temperature of the morning hemisphere is T [?] D T/2, and that of the evening hemisphere T + DT/2. The radiation reaction upon a surface element dA, normal to its surface, is dF = 2sT [sup.4] dA/3c. For a spherical particle of radius R, the Yarkovski force in the orbit plane due to the excess emission on the evening side is
F [sub.Y] = 8/3 p R [sup.2] cos ps. (52)
where s is the Stefan-Boltzmann constant and ps is the particle's obliquity, that is, the angle between its rotation axis and orbit pole. The reaction force is positive for an object that rotates in the prograde direction, 0 < ps < 90deg, and negative for an object with retrograde rotation, 90deg < ps < 180deg. In the latter case, the force enhances the Poynting-Robertson drag.
The Yarkovski force is important for bodies ranging in size from meters to several kilometers. Asymmetric out-gassing from comets produces a nongravitational force similar in form to the Yarkovski force. [See COMETARY DYNAMICS.]
7.4 Gas Drag
Although interplanetary space generally can be considered an excellent vacuum, there are certain situations in planetary dynamics where interactions with gas can significantly alter the motion of solid particles. Two prominent examples of this process are planetesimal interactions with the gaseous component of the protoplanetary disk during the formation of the solar system and orbital decay of ring particles as a result of drag caused by extended planetary atmospheres.
In the laboratory, gas drag slows solid objects down until their positions remain fixed relative to the gas. In the planetary dynamics case, the situation is more complicated. For example, a body on a circular orbit about a planet loses mechanical energy as a result of drag with a static atmosphere, but this energy loss leads to a decrease in semimajor axis of the orbit, which implies that the body actually speeds up! Other, more intuitive effects of gas drag are the damping of eccentricities and, in the case where there is a preferred plane in which the gas density is the greatest, the damping of inclinations relative to this plane.
Objects whose dimensions are larger than the mean free path of the gas molecules experience Stokes' drag,
F [sub.D] = [?], (53)
where v is the relative velocity of the gas and the body, r is the gas density, A is the projected surface area of the body, and C [sub.D] is a dimensionless drag coefficient, which is of order unity unless the Reynolds number is very small. Smaller bodies are subject to Epstein drag,
F [sub.D] = [?]Arvv' (54)
where v' is the mean thermal velocity of the gas. Note that as the drag force is proportional to surface area and the gravitational force is proportional to volume (for constant particle density), gas drag is usually most important for the dynamics of small bodies.
The gaseous component of the protoplanetary disk in the early solar system is believed to have been partially supported against the gravity of the Sun by a negative pressure gradient in the radial direction. Thus, less centripetal force was required to complete the balance, and consequently the gas orbited less rapidly than the Keplerian velocity. The "effective gravity" felt by the gas is
g[sub.eff] = [?] [?] (1/r) . (55)
To maintain a circular orbit, the effective gravity must be balanced by centripetal acceleration, rn [sup.2]. For estimated protoplanetary disk parameters, the gas rotated [?]0.5% slower than the Keplerian speed.
Large particles moving at (nearly) the Keplerian speed thus encountered a headwind, which removed part of their angular momentum and caused them to spiral inward toward the Sun. Inward drift was greatest for mid-sized particles, which have large ratios of surface area to mass yet still orbit with nearly Keplerian velocities. The effect diminishes for very small particles, which are so strongly coupled to the gas that the headwind they encounter is very slow. Peak rates of inward drift occur for particles that collide with roughly their own mass of gas in one orbital period. Meter-sized bodies in the inner solar nebula drift inward at a rate of up to 10[sup.6] km/yr! Thus, the material that survives to form the planets must complete the transition from centimeter to kilometer size rather quickly, unless it is confined to a thin dust-dominated subdisk in which the gas is dragged along at essentially the Keplerian velocity.
Drag induced by a planetary atmosphere is even more effective for a given density, as atmospheres are almost entirely pressure supported, so the relative velocity between the gas and particles is high. As atmospheric densities drop rapidly with height, particles decay slowly at first, but as they reach lower altitudes, their decay can become very rapid. Gas drag is the principal cause of orbital decay of artificial satellites in low Earth orbit.
7.5 Tidal Interactions and Planetary Satellites
Tidal forces are important to many aspects of the structure and evolution of planetary bodies:
1. On short timescales, temporal variations in tides (as seen in the frame rotating with the body under consideration) cause stresses that can move fluids with respect to more rigid parts of the planet (e.g., the familiar ocean tides) and even cause seismic disturbances (though the evidence that the Moon causes some earthquakes is weak and disputable, it is clear that the tides raised by Earth are a major cause of moonquakes).
2. On long timescales, tides cause changes in the orbital and spin properties of planets and moons. Tides also determine the equilibrium shape of a body located near any massive body; note that many materials that behave as solids on human timescales are effectively fluids on very long geological timescales (e.g., Earth's mantle).
The gravitational attraction of the Moon and Earth on each other causes tidal bulges that rise in a direction close to the line joining the centers of the two bodies. Particles on the nearside of the body experience gravitational forces from the other body that exceed the centrifugal force of the mutual orbit, whereas particles on the far side experience gravitational forces that are less than the centripetal forces needed for motion in a circle. It is the gradient of the gravitational force across the body that gives rise to the double tidal bulge.
The Moon spins once per orbit, so that the same face of the Moon always points toward Earth and the Moon is always elongated in that direction. Earth, however, rotates much faster than the Earth-Moon orbital period. Thus, different parts of Earth point toward the Moon and are tidally stretched. If the Earth was perfectly fluid, the tidal bulges would respond immediately to the varying force, but the finite response time of Earth's figure causes the tidal bulge to lag behind, at the point on Earth where the Moon was overhead slightly earlier. Since Earth rotates faster than the Moon orbits, this "tidal lag" on Earth leads the position of the Moon in inertial space. As a result, the tidal bulge of Earth accelerates the Moon in its orbit. This causes the Moon to slowly spiral outward. The Moon slows down Earth's rotation by pulling back on the tidal bulge, so the angular momentum in the system is conserved. This same phenomenon has caused most, if not all, major moons to be in synchronous rotation: the rotation and orbital periods of these bodies are equal. In the case of the Pluto-Charon system, the entire system is locked in a synchronous rotation and revolution of 6.4 days. Satellites in retrograde orbits (e.g., Triton) or satellites whose orbital periods are less than the planet's rotation period (e.g., Phobos) spiral inward toward the planet as a result of tidal forces.
Mercury orbits the Sun in 88 days and rotates around its axis in 59 days, a 3:2 spin-orbit resonance. Hence, at every perihelion one of two locations is pointed at the Sun: the subsolar longitude is either 0deg or 180deg. This configuration is stable because Mercury has both a large orbital eccentricity and a significant permanent deformation that is aligned with the solar direction at perihelion. Indeed, at 0deg longitude there is a large impact crater, Caloris Planitia, which may be the cause of the permanent deformation.
3. Under special circumstances, strong tides can have significant effects on the physical structure of bodies. Generally, the strongest tidal forces felt by solar system bodies (other than Sun-grazing or planet-grazing comets) are those caused by planets on their closest satellites. Near a planet, tides are so strong that they rip a fluid (or weakly aggregated solid) body apart. In such a region, large moons are unstable, and even small moons, which could be held together by material strength, are unable to accrete because of tides. The boundary of this region is known as Roche's limit. Inside Roche's limit, solid material remains in the form of small bodies and rings are found instead of large moons.
The closer a moon is to a planet, the stronger is the tidal force to which it is subjected. Let us consider Roche's limit for a spherical satellite in synchronous rotation at a distance r from a planet. This is the distance at which a loose particle on an equatorial subplanet point just remains gravitationally bound to the satellite. At the center of the satellite of mass m and radius R [sub.s], a particle would be in equilibrium and so
= n [sup.2] r, (56)
where M([?] m) is the mass of the planet. However, at the equator, the particle will experience (i) an excess gravitational or centrifugal force due to the planet, (ii) a centrifugal force due to rotation, and (iii) a gravitational force due to the satellite. If the equatorial particle is just in equilibrium, these forces will balance and
[?] R [sub.s] + n [sup.2] r = . (57)
In this case, Roche's limit r [sub.Roche] is given by
r [sub.Roche] = 3[sup.1/3] R [sub.planet]. (58)
with r[sub.planet] and r[sub.s] are the densities for the planet and satellite, respectively, and R [sub.planet] is the planetary radius. When a fluid moon is considered and flattening of the object due to the tidal distortion is taken into account, the correct result for a liquid moon (no internal strength) is
r [sub.Roche] = 2.456 R [sub.planet]. (59)
Most bodies have significant internal strength, which allows bodies with sizes [?][?]100 km to be stable somewhat inside Roche's limit. Mars's satellite Phobos is well inside Roche's limit; it is subjected to a tidal force equivalent to that in Saturn's B ring.
4. Internal stresses caused by variations in tides on a body in an eccentric orbit or not rotating synchronously with its orbital period can result in significant tidal heating of some bodies, most notably in Jupiter's moon Io. If no other forces were present, this would lead to a decay of Io's orbital eccentricity. By analogy to the Earth-Moon system, the tide raised on Jupiter by Io will cause Io to spiral outward and its orbital eccentricity to decrease. However, there exists a 2:1 mean-motion resonant lock between Io and Europa. Io passes on some of the orbital energy and angular momentum that it receives from Jupiter to Europa, and Io's eccentricity is increased as a result of this transfer. This forced eccentricity maintains a high tidal dissipation rate and large internal heating in Io, which displays itself in the form of active volcanism. [See IO]
7.6 Tidal Evolution and Resonances
Objects in prograde orbits that lie outside the synchronous orbit can evolve outward at different rates, so there may have been occasions in the past when pairs of satellites evolved toward an orbit-orbit resonance. The outcome of such a resonant encounter depends on the direction from which the resonance is approached. For example, capture into resonance is possible only if the satellites are approaching one another. If the satellites are receding, then capture is not possible, but the resonance passage can lead to an increase in the eccentricity and inclination. In certain circumstances it is possible to study the process using a simple mathematical model. However, this model breaks down near the chaotic separatrices of resonances and in regions of resonance overlap.
It is likely that the major satellites of Jupiter, Saturn, and Uranus have undergone significant tidal evolution and that the numerous resonances in the Jovian and Saturnian systems are a result of resonant capture. The absence of orbit-orbit resonances among the major moons in the Uranian system is thought to be related to the fact that the oblateness of Uranus is significantly less than that of Jupiter or Saturn. In these circumstances, there can be large chaotic regions associated with resonances and stable capture may be impossible. However, temporary capture into some resonances can produce large changes in eccentricity or inclination. For example, the Uranian satellite Miranda has an anomalously large inclination of 4deg, which is thought to be the result of a chaotic passage through the 3:1 resonance with Umbriel at some time in its orbital history. Under tidal forces, a satellite's eccentricity is reduced on a shorter timescale than its inclination, and Miranda's current inclination agrees with estimates derived from a chaotic evolution. [See PLANETARY SATELLITES.]
8. Chaotic Rotation
8.1 Spin-Orbit Resonance
One of the dissipative effects of the tide raised on a natural satellite by a planet is to cause the satellite to evolve toward a state of synchronous rotation, where the rotational period of the satellite is approximately equal to its orbital period. Such a state is one example of a spin-orbit resonance, where the ratio of the spin period to the orbital period is close to a rational number. The time needed for a near-spherical satellite to achieve this state depends on its mass and orbital distance from the planet. Small, distant satellites take a longer time to evolve into the synchronous state than do large satellites that orbit close to the planet. Observations by spacecraft and ground-based instruments suggest that most regular satellites are in the synchronous spin state, in agreement with theoretical predictions.
The lowest energy state of a satellite in synchronous rotation has the moon's longest axis pointing in the approximate direction of the planet-satellite line. Let th denote the angle between the long axis and the planet-satellite line in the planar case of a rotating satellite (Fig. 21). The variation of th with time can be described by equating the time variation of the rotational angular momentum with the restoring torque. The resulting differential equation is
tḧ + sin 2(th [?] f) = 0, (60)
where o[sub.0] is a function of the principal moments of inertia of the satellite, r is the radial distance of the satellite from the planet, and f is the true anomaly (or angular position) of the satellite in its orbit. The radius is an implicit function of time and is related to the true anomaly by the
FIGURE 21 The geometry used to define the orientation of a satellite in orbit about a planet. The planet-satellite line makes an angle f (the true anomaly) with a reference line, which is taken to be the periapse direction of the satellite's orbit. The orientation angle, th, of the satellite is the angle between its long axis and the reference direction.
r = , (61)
where a and e are the constant semimajor axis and the eccentricity of the satellite's orbit, respectively, and the orbit is taken to be fixed in space.
Equation (60) defines a deterministic system where the initial values of th and tḣ determine the subsequent rotation of the satellite. Since th and tḣ define a unique spin position of the satellite, a surface of section plot of (th, tḣ) once every orbital period, say at every periapse passage, produces a picture of the phase space. Figure 22 shows the resulting surface of section plots for a number of starting conditions using e = 0.1 and o[sub.0] = 0.2. The chosen values of o[sub.0] and e are larger than those that are typical for natural satellites, but they serve to illustrate the structure of the surface of section; large values of e are unusual since tidal forces also act to damp eccentricity. The surface of section shows large, regular regions surrounding narrow islands associated with the 1:2, 1:1, 3:2, 2:1, and 5:2 spin-orbit resonances at tḣ = 0.5, 1, 1.5, 2, and 2.5, respectively. The largest island is associated with the strong 1:1 resonance and, although other spin states are possible, most regular satellites, including Earth's Moon, are observed to be in this state. Note the presence of diffuse collections of points associated with small chaotic regions at the separatrices of the resonances. These are particularly obvious at the 1:1 spin-orbit state at th = p/2, tḣ = 1. Although this is a completely different dynamical system compared to the circular restricted three-body problem, there are distinct similarities in the types of behavior visible in Fig. 22 and parts of Figs. 14 and 15.
FIGURE 22 Representative surface of section plots of the orientation angle, th, and its time derivative, tḣ, obtained from the numerical solution of Eq. (59) using e = 0.1 and o[sub.0] = 0.2. The values of th and tḣ were obtained at every periapse passage of the satellite. Four starting conditions were integrated for each of the 1:2, 1:1, 3:2, 2:1, and 5:2 spin-orbit resonances in order to illustrate motion inside, at the separatrix, and on either side of each resonance. The thickest "island" is associated with the strong 1:1 spin-orbit state th = 1, whereas the thinnest is associated with the weak 5:2 resonance at th = 2.5.
In the case of near-spherical objects, it is possible to investigate the dynamics of spin-orbit coupling using analytical techniques. The sizes of the islands shown in Fig. 22 can be estimated by expanding the second term in Eq. (60) and isolating the terms that will dominate at each resonance. Using such a method, each resonance can be treated in isolation and the gravitational effects of nearby resonances can be neglected. However, if a satellite is distinctly nonspherical, o[sub.0] can be large and this approximation is no longer valid. In such cases it is necessary to investigate the motion of the satellite using numerical techniques.
Hyperion is a satellite of Saturn that has an unusual shape (Fig. 23). It has a mean radius of 135 km, an orbital eccentricity of 0.1, a semimajor axis of 24.55 Saturn radii, and a corresponding orbital period of 21.3 days. Such a small object at this distance from Saturn has a large tidal despinning timescale, but the unusual shape implies an estimated value of o[sub.0] = 0.89.
The surface of section for a single trajectory is shown in Fig. 24 using the same scale as Fig. 22. It is clear that there is a large chaotic zone that encompasses most of the spin-orbit resonances. The islands associated with the synchronous and other resonances survive but in a much reduced form. Although this calculation assumes that Hyperion's spin axis remains perpendicular to its orbital plane, studies have shown that the satellite should also be undergoing
FIGURE 23 Two Cassini images of the Saturnian satellite Hyperion show the unusual shape of the satellite, which is one cause of its chaotic rotation. Panel (a) is a true color image, while panel (b) uses false color and has better resolution because it was obtained at closer range. [Courtesy of NASA/JPL/Space Science Institute.]
a tumbling motion, such that its axis of rotation is not fixed in space.
Voyager observations of Hyperion indicated a spin period of 13 days, which suggested that the satellite was not in synchronous rotation. However, the standard techniques that are used to determine the period are not applicable if it varies on a timescale that is short compared with the times-pan of the observations. In principle, the rotational period can be deduced from ground-based observations by looking
FIGURE 24 A single surface of section plot of the orientation angle, th, and its time derivative, tḣ, obtained from the numerical solution of Eq. (10) using the values e = 0.1 and o[sub.0] = 0.89, which are appropriate for Hyperion. The points cover a much larger region of the phase space than any of those shown in Fig. 22, and although there are some remaining islands of stability, most of the phase space is chaotic.
for periodicities in plots of the brightness of the object as a function of time (the lightcurve of the object). The results of one such study for Hyperion are shown in Fig. 25. Since there is no recognizable periodicity, the lightcurve is consistent with that of an object undergoing chaotic rotation. Hyperion is the first natural satellite that has been observed to have a chaotic spin state, and results from Cassini images confirm this result. Observations and numerical studies of Hyperion's rotation in three dimensions have shown that its
FIGURE 25 Ground-based observations by J. Klavetter of Hyperion's lightcurve obtained over 13 weeks (4.5 orbital periods) in 1987. The fact that there is no obvious curve through the data points is convincing evidence that the rotation of Hyperion is chaotic. (Courtesy of the American Astronomical Society.)
spin axis does not point in a fixed direction. Therefore the satellite also undergoes a tumbling motion in addition to its chaotic rotation.
The dynamics of Hyperion's motion is complicated by the fact that it is in a 4:3 orbit-orbit resonance with the larger Saturnian satellite Titan. Although tides act to decrease the eccentricities of satellite orbits, Hyperion's eccentricity is maintained at 0.104 by means of the resonance. Titan effectively forces Hyperion to have this large value of e and so the apparently regular orbital motion inside the resonance results, in part, in the extent of the chaos in its rotational motion. [See PLANETARY SATELLITES.]
8.3 Other Satellites
Although there is no evidence that other natural satellites are undergoing chaotic rotation at the present time, it is possible that several irregularly shaped regular satellites did experience chaotic rotation at some time in their histories. In particular, since satellites have to cross chaotic separatrices before capture into synchronous rotation can occur, they must have experienced some episode of chaotic rotation. This may also have occurred if the satellite suffered a large impact that affected its rotation. Such episodes could have induced significant internal heating and resurfacing events in some satellites. The Martian moon Phobos and the Uranian moon Miranda have been mentioned as possible candidates for this process. If this happened early in the history of the solar system, then the evidence may well have been obliterated by subsequent cratering events. [See PLANETARY SATELLITES.]
8.4 Chaotic Obliquity
The fact that a planet is not a perfect sphere means that it experiences additional perturbing effects due to the gravitational forces exerted by its satellites and the Sun, and these can cause long-term evolution in its obliquity (the angle between the planet's equator and its orbit plane). Numerical investigations have shown that chaotic changes in obliquity are particularly common in the inner solar system. For example, it is now known that the stabilizing effect of the Moon results in a variation of +- 1.3deg in Earth's obliquity around a mean value of 23.3deg. Without the Moon, Earth's obliquity would undergo large, chaotic variations. In the case of Mars there is no stabilizing factor and the obliquity varies chaotically from 0deg to 60deg on a timescale of 50 million years. Therefore an understanding of the long-term changes in a planet's climate can be achieved only by an appreciation of the role of chaos in its dynamical evolution.
It is clear that nonlinear dynamics has provided us with a deeper understanding of the dynamical processes that have helped to shape the solar system. Chaotic motion is a natural consequence of even the simplest systems of three or more interacting bodies. The realization that chaos has played a fundamental role in the dynamical evolution of the solar system came about because of contemporary and complementary advances in mathematical techniques and digital computers. This coincided with an explosion in our knowledge of the solar system and its major and minor members. Understanding how a random system of planets, satellites, ring and dust particles, asteroids, and comets interacts and evolves under a variety of chaotic processes and timescales, ultimately means that this knowledge can be used to trace the history and predict the fate of other planetary systems.
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|Publication:||Encyclopedia of the Solar System, 2nd ed.|
|Article Type:||Topic overview|
|Date:||Jan 1, 2007|
|Next Article:||Planetary Volcanism.|