SP-345 Evolution of the Solar System





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 is equivalent toT change the circular orbit into an elliptical orbit with eccentricity e. Axial oscillations (perpendicular to the preferred plane), with a period approximately
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.).



The coordinate system adopted in subsequent discussions is the modified spherical coordinate system with [Greek letter] phi, [Greek letter] lambda, 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


C = rorb x vorb


where rorb is the orbital distance and vorb 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 fG (per unit mass) of the central body and by the centrifugal force fc (per unit mass)


mathematical equation(3.2.2)

where mathematical symbol
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 fG is exactly balanced by the centrifugal force. We have

mathematical equation(3.2.3)

The orbital angular velocity is


mathematical equation


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



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 +deltar, it is acted upon by the force

mathematical equation

Because the force is zero for r = r0 we obtain

mathematical equation


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


mathematical equation


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


mathematical equation


The angular velocity of this axial oscillation is


mathematical equation



The guiding-center method of approximating the Kepler motion.

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 Mc 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 Mc is er0. The position of the pericenter is given by mathematical symbol. The difference between the undashed circle and the exact Kepler ellipse is really less than the thickness of the line.


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

mathematical equation

We place a moving coordinate system with the origin at a point traveling along the unperturbed (circular) orbit with the angular velocity mathematical symbol(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



mathematical equation


mathematical equation

[41] where [Greek letter] phi
, 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

mathematical equation

where mathematical symbol
and Kr are constants. Because C is constant, we have


mathematical equation


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


mathematical equation

where we have introduced

mathematical equation

We find


mathematical equation


or after integration


mathematical equation


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


mathematical equation

Assuming the pericenter mathematical symbol
to be

mathematical equation

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

In a similar way, we find the axial oscillations:


mathematical equation


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

mathematical equation

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


mathematical equation



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


mathematical equation


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


C = (GMcr0)1/2 (3.4.2)


From eq. (3.4.1) we find

mathematical equation

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


mathematical equation


where the Kepler angular velocity is


mathematical equation

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 mathematical equation
, 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"


mathematical equation


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


mathematical equation

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.



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



mathematical equation


where r0 is the radius of the unperturbed motion, defined by eq. (3.4.2) and mathematical equation
, 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


mathematical equation


mathematical equation


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



According to eq. (3.4.4), the motion in a 1/r2 field is degenerate, in the sense thatmathematical equation
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


mathematical equation


taking account only of the first term from the equatorial bulge. The constant [Greek letter] capital lambdais always positive. From eq. (3.6.1), we find


mathematical equation


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


mathematical equation


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


mathematical equation

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

mathematical equation

in the retrograde direction.

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


mathematical equation

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


mathematical equation


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


mathematical equation




mathematical equation


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



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.