Solution of the Equation of Motion

The equation of motion (6.5) is a second-order (i. e. contains second derivatives) vector valued differential equation. Therefore we need six integration constants or integrals for the complete solution. The solution is an infinite family of orbits with different sizes, shapes and orientations. A particular solution (e. g. the orbit of Jupiter) is selected by fixing the values of the six integrals. The fate of a planet is unambiguously determined by its position and velocity at any given moment; thus we could take the position and velocity vectors at some moment as our integrals. Although they do not tell us anything about the geometry of the orbit, they can be used as initial values when integrating the orbit numerically with a computer. Another set of integrals, the orbital elements, contains geometric quantities describing the orbit in a very clear and concrete way. We shall return to these later. A third possible set involves certain physical quantities, which we shall derive next.

We begin by showing that the angular momentum remains constant. The angular momentum of the planet in the heliocentric frame is

k = r x r. Let us find the time derivative of this: k = r x r+r x r.

The latter term vanishes as a vector product of two parallel vectors. The former term contains r, which is given by the equation of motion:

Thus k is a constant vector independent of time (as is L, of course).

Since the angular momentum vector is always perpendicular to the motion (this follows from (6.8)), the motion is at all times restricted to the invariable plane perpendicular to k (Fig. 6.2).

To find another constant vector, we compute the vector product k x r:

The time derivative of the distance r is equal to the projection of r in the direction of r (Fig. 6.3); thus, using the properties of the scalar product, we get r = r ■ r/r, which gives r ■ r = rr . Hence,

The vector product can also be expressed as k x r = — (k x r) , dt( )

Celestial mechanicians usually prefer to use the angular momentum divided by the planet's mass

Fig. 6.2. The angular momentum vector k is perpendicular to the radius and velocity vectors of the planet. Since k is a constant vector, the motion of the planet is restricted to the plane perpendicular to k

6.2 Solution of the Equation of Motion

Fig. 6.3. The radial velocity r is the projection of the velocity vector r in the direction of the radius vector r

since k is a constant vector. Combining this with the previous equation, we have d dt and

Since k is perpendicular to the orbital plane, k x r must lie in that plane. Thus, e is a linear combination of two vectors in the orbital plane; so e itself must be in the orbital plane (Fig. 6.4). Later we shall see that it points to the direction where the planet is closest to the Sun in its orbit. This point is called the perihelion.

One more constant is found by computing r ■ r:

we get d

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