Rotation of the Moon and Hyperion

The approximation to the potential energy that we have derived can be used for a number of different problems. For instance, it can be used to investigate the effect of oblateness on the evolution of an artificial satellite about the Earth, or to incorporate the effect of planetary oblateness on the evolution of the orbits of natural satellites, such as the Moon, or the Galilean satellites of Jupiter. However, as the principal application here, we will use it to investigate the rotational dynamics of natural satellites and planets.

The potential energy depends on the position of the point mass relative to the rigid body and on the orientation of the rigid body.

Thus the changing orientation is coupled to the orbital evolution;

each affects the other. However, in many situations the effect of the orientation of the body on the evolution of the orbit may be ignored. One way to see this is to look at the relative magnitudes of the two terms in the potential energy (2.74). We already know that the second term is guaranteed to be smaller than the first by a factor of (ξ_max/R)², but often it is much smaller still because the body involved is nearly spherical. For example, the radius of the Moon is about a third the radius of the Earth and the distance to the Moon is about 60 Earth-radii. So the second term is smaller than the first by a factor of order 10⁻⁴ due to the size factors. In addition the Moon is roughly spherical and for any orientation the combination A+B+C−3I is of order 10⁻⁴C. Now C is itself of order ²₅M R², because the density of the Moon does not vary strongly with radius. So for the Moon the second term is of order 10⁻⁸ relative to the first. Even radical changes in the orientation of the Moon would have little dynamical effect on the orbit of the Moon.

We can learn some important qualitative aspects of the orien-tation dynamics by studying a simplified model problem. First, we assume that the body is rotating about its largest moment of inertia. This is a natural assumption. Remember that for a free

rigid body the loss of energy while conserving angular momentum leads to rotation about the largest moment of inertia. This is observed for most bodies in the solar system. Next, we assume that the spin axis is perpendicular to the orbital motion. This is a good approximation for the rotation of natural satellites, and is a natural consequence of tidal friction—dissipative solid body tides raised on the satellite by the gravitational interaction with the planet. Finally, for simplicity we take the rigid body to be mov-ing on a fixed elliptic orbit. This may approximate the motion of some physical systems, provided the timescale of the evolution of the orbit is large compared to any timescale associated with the rotational dynamics that we are investigating. So we have a nice toy problem. This problem has been used to investigate the rotational dynamics of Mercury, the Moon, and other natural satellites. It makes specific predictions concerning the rotation of Phobos, a satellite of Mars, which can be compared with observa-tions. It provides a basic understanding of the fact that Mercury rotates precisely 3 times for every 2 orbits it completes, and is the starting point for understanding the chaotic tumbling of Saturn’s satellite Hyperion.

$ ˆ

a

f θ

ˆb θ_a θ_b

Figure 2.10 The spin-orbit model problem in which the spin axis is constrained to be perpendicular to the orbit plane has a single degree of freedom, the orientation of the body in the orbit plane. Here the orientation is specified by the generalized coordinateθ.

We are assuming that the orbit does not change or precess. The orbit is an ellipse with the point mass at a focus of the ellipse. The angle f (see figure 2.10) measures the position of the rigid body in its orbit relative to the point in the orbit at which the two bodies are closest.¹⁸ We assume the orbit is a fixed ellipse, so the angle f and the distance R are periodic functions of time, with period equal to the orbit period. With the spin axis constrained to be perpendicular to the orbit plane, the orientation of the rigid body is specified by a single degree of freedom: the orientation of the body about the spin axis. We specify this orientation by the generalized coordinateθthat measures the angle to the ˆaprincipal axis from the same line as we measuref, the line through the point of closest approach.

Having specified the coordinate system, we can work out the details of the kinetic and potential energies, and thus find the Lagrangian. The kinetic energy is

T(t, θ,θ) =˙ ¹₂Cθ˙², (2.75)

where C is the moment of inertia about the spin axis, and the angular velocity of the body about the ˆc axis is ˙θ. There is no component of angular velocity on the other principal axes.

To get an explicit expression for the potential energy we must write the direction cosines in terms of θ and f: α = cosθ_a =

−cos(θ−f),β = cosθ_b = sin(θ−f), and γ = cosθ_c = 0 because the ˆcaxis is perpendicular to the orbit plane. The potential energy is then

−GM M⁰ R

−1 2

GM⁰ R³

£(1−3 cos²(θ−f))A+ (1−3 sin²(θ−f))B+C¤ . Since we are assuming that the orbit is given, we only need to keep terms that depend onθ. Expanding the squares of the cosine and the sine in terms of the double angles, and dropping all the

18Traditionally, the point in the orbit at which the two bodies are closest is called thepericenter, and the anglefis called thetrue anomaly.

terms that do not depend on θ we find the potential energy for A Lagrangian for the model spin-orbit coupling problem is then L=T−V: We introduce the dimensionless “out-of-roundness” parameter

²=

r3(B−A)

C , (2.78)

and use the fact that the orbit frequencynsatisfies Kepler’s third lawn²a³ =G(M+M⁰), which is approximatelyn²a³ =GM⁰for a small body in orbit around a much more massive one (M ¿M⁰).

In terms of ²and nthe spin-orbit Lagrangian is L(t, θ,θ) =˙ 1

2Cθ˙²+n²²²C 4

a³

R³(t)cos 2(θ−f(t)). (2.79) This is a problem with one degree of freedom with terms that vary periodically with time.

The Lagrange equations are derived in the usual manner. The equations are

CD²θ(t) =−n²²²C 2

a³

R³(t)sin 2(θ(t)−f(t)). (2.80) The equation of motion is very similar to that of the periodically driven pendulum. The main difference here is that not only is the strength of the acceleration changing periodically, but in the spin-orbit problem the center of attraction is also varying periodically.

We can give a physical interpretation of this equation of motion.

It states that the rate of change of the angular momentum is equal to the applied torque. The torque on the body arises because the

19The given potential energy differs from the actual potential energy in that non-constant terms that do not depend onθ and consequently do not affect the evolution ofθ have been dropped.

body is out of round and the gravitational force varies as the inverse square of the distance. Thus the force per unit mass on the near side of the body is stronger than the acceleration of the body as a whole, and the force per unit mass on the far side of the body is a little less than the acceleration of the body as a whole. Thus, relative to the acceleration of the body as a whole the far side is forced outward while the inner part of the body is forced inward. The net effect is a torque on the body, which tries to align the long axis of the body with the line to the external point mass. If θ is a bit larger than f then there is a negative torque, and if θ is a bit smaller than f then there is a positive torque, both of which would align the long axis with the planet if given a fair chance. The torque arises because of the difference of the inverseR² force across the body, so the torque is proportional to R⁻³. There is only a torque if the body is out-of-round, for otherwise there is no handle to pull on. This is reflected in the factorB−A, which appears in the expression for the torque. The potential depends only on the moment of inertia, thus the body has the same dynamics if it is rotated by 180^◦. The factor of 2 in the argument of sine reflects this symmetry. This torque is called the “gravity gradient torque.”

To compute the evolution requires a bunch of detailed prepara-tion similar to what has been done for other problems. There are many interesting phenomena to explore. We can take parameters appropriate for the Moon, and find that Mr. Moon does not con-stantly point the same face to the Earth, but instead concon-stantly shakes his head in dismay at what goes on here. If we nudge the Moon a bit, say by hitting it with an asteroid, we find that the long axis oscillates back and forth with respect to the direction that points to the Earth. For the Moon, the orbital eccentric-ity is currently about 0.05, and the out-of-roundness parameter is about ²= 0.026. Figure 2.11 shows the angleθ−f as a function of time for two different values of the “lunar” eccentricity. The plot spans 50 lunar orbits, or a little under 4 years. This Moon has been kicked by a large asteroid and has initial rotational angu-lar velocity ˙θequal to 1.01 times the orbit frequency. The initial orientation is θ = 0. The smooth trace shows the evolution if the orbital eccentricity is set to zero. We see an oscillation with a period of about 40 lunar orbit periods or about 3 years. The more wiggly trace shows the evolution of θ−f with an orbital eccentricity of 0.05, near the current lunar eccentricity. The lunar

50 25

0 1

0

−1

Figure 2.11 The angle θ−f versus time for 50 orbit periods. The ordinate scale is±1 radian. The Moon has been kicked so that the initial rotational angular velocity is 1.01 times the orbital frequency. The trace with fewer wiggles was computed with zero lunar orbital eccentricity;

the other trace was computed with lunar orbital eccentricity of 0.05.

The period of the rapid oscillations is the lunar orbit period, and are due mostly to the nonuniform motion off.

eccentricity superimposes an apparent shaking of the face of the moon back and forth with the period of the lunar orbit. Though the Moon does slightly change its rate of rotation during the course of its orbit, most of this shaking is due to the nonuniform motion of the Moon in its elliptical orbit. This oscillation is called the

“optical libration of the Moon,” and it allows us to see a bit more than half the surface of the Moon. The longer period oscillation induced by the kick is called the “free libration of the Moon.” It is “free” because we are free to excite it by choosing appropriate initial conditions. The mismatch of the orientation of the moon caused by the optical libration actually produces a periodic torque on the Moon, which slightly speeds up and slows down the Moon during every orbit. The resulting oscillation is called the “forced libration of the Moon,” but it is too small to see in this plot.

The oscillation period of the free libration is easily calculated.

We see that the eccentricity of the orbit does not substantially

affect the period, so consider the special case of zero eccentricity.

In this caseR=a, a constant, andf(t) =ntwherenis the orbital frequency (traditionally called the mean motion). The equation of motion becomes

D²θ(t) =−n²²²

2 sin 2(θ(t)−nt). (2.81)

Let ϕ(t) = θ(t)−nt, and consequently Dϕ(t) = Dθ(t)−n, and D²ϕ=D²θ. Substituting these, the equation governing the evo-lution of ϕis

D²ϕ=−n²²²

2 sin 2ϕ. (2.82)

For small deviations from synchronous rotation (small ϕ) this is

D²ϕ=−n²²²ϕ, (2.83)

so we see that the small amplitude oscillation frequency of ϕ is n². For the Moon,²is about 0.026, so the period is about 1/0.026 orbit periods or about 40 lunar orbit periods, which is what we observed.

It is perhaps more fun to see what happens if the out-of-roundness parameter is large. After our experience with the driven pendulum it is no surprise that we find abundant chaos in the spin-orbit problem when the system is strongly driven by having large

²and significant e. There is indeed one body in the solar system that exhibits chaotic rotation—Hyperion, a small satellite of Sat-urn. Though our model is not adequate for a complete account of Hyperion, we can show that our toy model exhibits chaotic be-havior for parameters appropriate for Hyperion. We take²= 0.89 and e= 0.1. Figure 2.12 shows θ−f for 50 orbits, starting with θ= 0 and ˙θ= 1.05. We see that sometimes one face of the body oscillates facing the planet, sometimes the other face oscillates facing the planet, and sometimes the body rotates relative to the planet in either direction.

If we were to relax our restriction that the spin axis is fixed per-pendicular to the orbit, then we find that the Moon maintains this orientation of the spin axis even if nudged a bit, but for Hyperion the spin axis almost immediately falls away from this configura-tion. The state in which Hyperion on average points one face to

50 25

0 π

0

−π

Figure 2.12 The angle θ−f versus time for 50 orbit periods. The ordinate scale is ±π radian. The out-of-roundness parameter is large

²= 0.89, with an orbital eccentricity ofe= 0.1. The system is strongly driven. The rotation is apparently chaotic.

Saturn is dynamically unstable to chaotic tumbling. Observations of Hyperion have confirmed that Hyperion is chaotically tumbling.