[**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

The body is acted upon by the specific
gravitational attraction **f _{G}** (per unit mass)
of the central body and by the centrifugal force

where is the tangential velocity component.

The simplest type of motion is that motion
with constant orbital velocity **v _{0}** in a circle with
radius

The orbital angular velocity is

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
**r _{0}** to

Because the force is zero for **r** = **r _{0}** we obtain

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

If the body is displaced in the
**z**
direction (axial direction), it is acted upon by the force
**f _{z}** which, because
div

The angular velocity of this axial oscillation is

[

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

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

and

[**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
**er _{0}**
(<<

where and **K _{r}**

As **x** <<
**r _{0}** and

where we have introduced

We find

or after integration

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

Assuming the pericenter to be

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:

where **i** (<<1) is the
inclination, **K _{z}** is a constant
and

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

** **

**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

[**43**] where
**M _{c}** is the mass of
the central body and

** **

From eq. (3.4.1) we find

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

where the Kepler angular velocity is

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"

The center of the epicycle moves with constant
velocity along the circle **r _{0}**. 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

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/**r**^{2} 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

where **r _{0}** 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

with

[**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/**r**^{2} 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/**r**^{2} 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/**r**^{2} 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*

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

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

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

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

in the retrograde direction.

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

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

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

where

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.