Evolution equations for two-planet solar system

In an inertial reference frame, the equations of motion of the various elements of our simplified solar system are

(10.1) | ||||||

(10.2) | ||||||

and | (10.3) |

It thus follows that

(10.4) | ||||||

and | (10.5) |

where and . The right-hand sides of these equations specify the interplanetary interaction forces that were neglected in our previous analysis. These right-hand sides can be conveniently expressed as the gradients of potentials:

where

with , and . Here, and are termed

In the absence of the second planet, the orbit of the first planet is fully described by its
six standard orbital elements (which are constants of its motion): the major radius,
; the mean longitude at epoch,
; the eccentricity,
;
the inclination (to the ecliptic plane),
; the longitude of the perihelion,
; and the longitude of the ascending node,
. (See Section 4.12.) As described in Appendix G, the perturbing influence of the second planet
causes these elements to slowly evolve in time. Such time-varying orbital elements are generally known as *osculating
elements*.^{10.1}
Actually, when describing the aforementioned evolution, it is more convenient to
work in terms of an alternative set of osculating elements, namely,
,
,
,
,
, and
. Here,
and
, where
is the unperturbed mean
orbital angular velocity. In the following, for ease of notation,
and
are written simply as
and
, respectively. Furthermore,
will be used as shorthand for
.
The evolution equations for the first planet's osculating orbital elements
take the form (see Section H.2)

where (see Section H.3)

Here, , , and

(10.19) |

where , , , , , are the osculating orbital elements of the second planet. The factors are known as

There is an analogous set of equations, which can be derived from Equations (10.7) and (10.9), that describe the time evolution of the osculating orbital elements of the second planet due to the perturbing influence of the first. These take the form (see Section H.2)

where (see Section H.3)

Here, , and .