[37] 3.1 THE GUIDING-CENTER APPROXIMATION OF CELESTIAL MECHANICS

The dynamic state of a celestial body can be represented by nine quantities. Of these, three give the position of the body (e.g., its center of gravity) at a certain moment, three give its three-dimensional velocity, and three give its spin (around three orthogonal axes). These quantities vary more or less rapidly in a way which can be found from the Nautical Almanac. In our study of the origin and the long-time evolution of the dynamic state of the solar system, we are predominantly interested in those dynamic quantities which are invariant or vary very slowly.

The typical orbits of satellites and planets are circles in certain preferred planes. For the satellite systems, the preferred planes tend to coincide with the equatorial planes of the central bodies. For the planetary system, the preferred plane is essentially the orbital plane of Jupiter (because this is the biggest planet), which is close to the plane of the ecliptic. The circular motion with period T is usually modified by superimposed oscillations. Radial oscillations (in the preferred plane) with period T change the circular orbit into an elliptical orbit with eccentricity e. Axial oscillations (perpendicular to the preferred plane), with a period T, give the orbit an inclination i to this plane.

With some exaggeration one may say that the goal of the traditional presentation of celestial mechanics was utility for the preparation of the Nautical Almanac and, more recently, for calculation of spacecraft trajectories. This approach is not very suitable if we want to study the mutual interaction between orbiting grains or the interaction of orbiting grains with a plasma or any viscous medium. It is more convenient to use an approximation method that treats an elliptical orbit as a perturbation of a circular orbit. This method is applicable only for orbits with small eccentricities. From a formal point of view the method has some similarity to the guiding-center method of treating the motion of charged particles in a magnetic field (Alfvén and Falthammar, 1963, p. 18 ff.).

[38] 3.2. CIRCULAR ORBITS

The coordinate system adopted in subsequent discussions is the modified spherical coordinate system with , , and r as the azimuthal angle or longitude, the meridional angle or latitude, and the radial direction, respectively. When rectangular coordinates are used the x-y plane lies in the equatorial plane and z is the axial direction.

For a body of negligible mass moving around a central body the specific angular momentum C (per unit mass) of the small body with reference to the central body (or, strictly speaking, to the center of gravity) is defined as

where r_orb is the orbital distance and v_orb is the orbital velocity of the small body. C is an invariant vector during the motion.

The body is acted upon by the specific gravitational attraction f_G (per unit mass) of the central body and by the centrifugal force f_c (per unit mass)

(3.2.2)

where is the tangential velocity component.

The simplest type of motion is that motion with constant orbital velocity v₀ in a circle with radius r₀. The gravitational force f_G is exactly balanced by the centrifugal force. We have

(3.2.3)

The orbital angular velocity is

(3.2.4)

The period of this motion is known as the Kepler period.

[39] 3.3. OSCILLATIONS MODIFYING THE CIRCULAR ORBIT

The circular orbit of the body can be modified by both radial and axial oscillations.

If the body is displaced radially from r₀ to r = r₀ +r, it is acted upon by the force

(3.3.1)

Because the force is zero for r = r₀ we obtain

(3.3.2)

As the angular frequency of a harmonic oscillator is , the body oscillates radially about the circle with

(3.3.3)

If the body is displaced in the z direction (axial direction), it is acted upon by the force f_z which, because div f = 0, is given by

(3.3.4)

The angular velocity of this axial oscillation is

(3.3.5)

Figure 3.3.1.- The guiding-center method of approximating the Kepler motion. The guiding center moves with constant velocity along the dashed circle of radius r₀ in the center of which the gravitating mass M_c is situated. The body M moves in an "epicycle" around the guiding center. The epicycle is an ellipse with the axis ratio 2/1 and semiminor axis of er₀. The epicycle motion is retrograde. The resulting motion of M is an ellipse which almost coincides with the undashed circle which has its center at O. The distance from O to M_c is er₀. The position of the pericenter is given by . The difference between the undashed circle and the exact Kepler ellipse is really less than the thickness of the line.

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From eqs. (3.2.4), (3.3.3) and (3.3.5)

(3.3.6)

We place a moving coordinate system with the origin at a point traveling along the unperturbed (circular) orbit with the angular velocity (fig. 3.3.1). The x axis points in the radial direction and the y axis in the forward tangential direction. The origin is called the "guiding center." We have

(3.3.7)

and

(3.3.8)

[41] where , is the angle measured from a fixed direction and t is counted from the moment when the guiding center is located in this fixed direction.

A radial harmonic oscillation with amplitude er₀ (<<r₀) can be written

(3.3.9)

where and K_r are constants. Because C is constant, we have

(3.3.10)

As x << r₀ and y << r₀ we find from eqs. (3.2.4) and (3.3.7-3.3.10):

(3.3.11)

where we have introduced

(3.3.12)

We find

(3.3.13)

or after integration

(3.3.14)

[42] The pericenter (point of nearest approach to the gravitating center) is reached when x is a minimum; that is, when

(3.3.15)

Assuming the pericenter to be

(3.3.16)

eq. (3.3.15) gives the expected periodicity of the pericenter, Thus, the pericenter moves (has a "precession") with the velocity, given by eq. (3.3.12).

In a similar way, we find the axial oscillations:

(3.3.17)

where i (<<1) is the inclination, K_z is a constant and

(3.3.18)

The angle of the "ascending node" (point where z becomes positive) is given by

(3.3.19)

3.4. MOTION IN AN INVERSE-SQUARE-LAW GRAVITATIONAL FIELD

If the mass of the orbiting body is taken as unity, then the specific gravitational force is

(3.4.1)

[43] where M_c is the mass of the central body and G is the gravitational constant. As f_c= f_G for the unperturbed motion, we have from eqs. (3.2.2) and (3.4.1)

From eq. (3.4.1) we find

(3.4.3)

Substituting eq. (3.4.3) into eqs. (3.3.3) and (3.3.5), eq. (3.3.6) reduces to

(3.4.4)

where the Kepler angular velocity is

(3.4.5)

The significance of eq. (3.4.4) is that, for the almost circular motion in an inverse-square-law field, the frequencies of radial and axial oscillation coincide with the fundamental angular velocity of circular motion. Consequently, we have , and there is no precession of the pericenter or of the nodes. According to eqs. (3.3.11) and (3.3.14), the body moves in the "epicycle"

(3.4.6)

The center of the epicycle moves with constant velocity along the circle r₀. The motion in the epicycle takes place in the retrograde direction. See fig. 3.3.1.

[44] Similarly, eq. (3.3.17) for the axial oscillation reduces to

(3.4.7)

We still have an ellipse, but its plane has the inclination i with the plane of the undisturbed circular motion. The axial oscillation simply means that the plane of the orbit is changed from the initial plane, which was arbitrarily chosen because in a 1/r² field there is no preferred plane.

3.5. NONHARMONIC OSCILLATION; LARGE ECCENTRICITY

If the amplitude of the oscillations becomes so large that the eccentricity is not negligible, the oscillations are no longer harmonic. This is the case for most comets and meteroids. It can be shown that instead of eq. (3.3.11) we have the more general formula

(3.5.1)

where r₀ is the radius of the unperturbed motion, defined by eq. (3.4.2) and , the angle between the vector radius of the orbiting body and of the pericenter of its orbit. The relation of eq. (3.4.4) is still valid, but the period becomes

(3.5.2)

with

(3.5.3)

[45] It can be shown that geometrically the orbit is an ellipse, with a the semimajor axis and e the eccentricity.

3.6. MOTION IN THE FIELD OF A ROTATING CENTRAL BODY

According to eq. (3.4.4), the motion in a 1/r² field is degenerate, in the sense that This is due to the fact that there is no preferred direction.

In the planetary system and in the satellite systems, the motions are perturbed because the gravitational fields deviate from pure 1/r² fields. This is essentially due to the effects discussed in this section and in sec. 3.7.

The axial rotations (spins) produce oblateness in the planets. We can consider their gravitation to consist of a 1/r² field from a sphere, on which is superimposed the field from the "equatorial bulge." The latter contains higher order terms but has the equatorial plane as the plane of symmetry. We can write the gravitational force in the equatorial plane

(3.6.1)

taking account only of the first term from the equatorial bulge. The constant is always positive. From eq. (3.6.1), we find

(3.6.2)

Substituting eq. (3.6.2), we have from eqs. (3.2.4), (3.3.3), and (3.3.5)

(3.6.3)

According to egs. (3.3.12) and (3.3.18), this means that the pericenter moves with the angular velocity

(3.6.4)

[46] in the prograde direction, and the nodes move with the angular velocity

(3.6.5)

in the retrograde direction.

Further, we obtain from eqs. (3.3.6), (3.6.4), and (3.6.5)

(3.6.6)

As the right-hand term is very small, we find to a first approximation

(3.6.7)

This is a well-known result in celestial mechanics. Using this last result in eq. (3.6.6) we find, to a second approximation,

(3.6.8)

where

(3.6.9)

A comparison of eq. (3.6.9) with calculations of by exact methods (Alfvén and Arrhenius, 1970a, p. 349) shows a satisfactory agreement.

3.7. PLANETARY MOTION PERTURBED BY OTHER PLANETS

The motion of the body we are considering is perturbed by other bodies orbiting in the same system. Except when the motions are commensurable [47] so that resonance effects become important, the main perturbation can be computed from the average potential produced by other bodies.

As most satellites are very small compared to their central bodies, the mutual perturbations are very small and of importance only in case of resonance. The effects due to planetary flattening described in sec. 3.6 dominate in the satellite systems. On the other hand, because the flattening of the Sun makes a negligible contribution, the perturbation of the planetary orbits is almost exclusively due to the gravitational force of the planets, among which the gravitational effect of Jupiter dominates. To calculate this to a first approximation, one smears out Jupiter's mass along its orbit and computes the gravitational potential from this massive ring. This massive ring would produce a perturbation which, both outside and inside Jupiter's orbit, would obey eq. (3.6.2). Hence eqs. (3.6.3)-(3.6.5) are also valid. The dominating term for the calculation of the perturbation of the Jovian orbit derives from a similar effect produced by Saturn. Where resonance effects occur (ch. 8), these methods are not applicable.

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