Expansion of planetary disturbing functions

where

(H.38) |

and

In the preceding equations, is the angle subtended between the directions of and .

Now,

(H.41) |

Let

(H.42) | ||||

(H.43) | ||||

and | (H.44) |

where and . Here, and are the true anomalies of the first and second planets, respectively. We expect and to both be [see Equation (H.54)], and to be [see Equation (H.64)]. We can write

(H.45) |

where

(H.46) |

Expanding in , and retaining terms only up to , we obtain

(H.47) |

where , and

(H.48) |

Hence,

(H.49) |

Now, we can expand and as Fourier series in :

(H.50) | ||

(H.51) |

where

(H.52) |

Incidentally, the are known as

where now denotes .

Equation (4.87) gives

to . Here, is the first planet's mean anomaly. Obviously, there is an analogous equation for . Moreover, from Equation (4.86), we have

(H.55) | ||

(H.56) |

Hence,

(H.57) |

and

(H.58) |

Thus, Equations (4.72)-(4.74) yield

(H.59) | ||||||

(H.60) | ||||||

and | (H.61) |

where is assumed to be . Here, , , are the Cartesian component of . There are, of course, completely analogous expressions for the Cartesian components of .

Now,

(H.62) |

so

It is easily demonstrated that represents the value taken by when . Hence, from Equations (H.44) and (H.63),

Now,

(H.65) |

However, from Equation (4.86),

(H.66) |

because . Likewise,

(H.67) |

Hence, we obtain

Equations (H.53), (H.54), (H.64), and (H.68) yield

where

Likewise, Equations (H.39), (H.40), (H.54), and (H.63) give

and

We can distinguish two different types of term that appear in our expansion
of the disturbing functions. *Periodic terms* vary sinusoidally
in time as our two planets orbit the Sun (i.e., they depend on the mean
ecliptic longitudes,
and
), whereas *secular terms* remain constant in time (i.e., they do not depend on
or
). We expect the periodic terms to give rise to relatively short-period (i.e., of order a typical orbital period) oscillations in the osculating orbital
elements of our planets. On the other hand, we expect the secular terms to
produce an initially linear increase in these elements with time. Because
such an increase can become significant over a long period of time, even
if the rate of increase is small, it is necessary to evaluate the secular terms
in the disturbing function to higher order than the periodic terms.
Hence, in the following, we shall evaluate periodic terms to
,
and secular terms to
.

Making use of Equations (H.20), (H.36), and (H.69)-(H.71), as well as the definitions (G.121)-(G.124) (with and ), the order by order expansion (in ) of the first planet's disturbing function becomes , where

Likewise, the order by order expansion of the second planet's disturbing function becomes , where