Expansion of Lagrange planetary equations

where , and is . Thus, because , the Lagrange planetary equations, Equations (G.125)-(G.130), applied to the first planet, reduce to

where

(H.8) | ||||||

and | (H.9) |

The Sun is much more massive than any planet in the solar system. It follows that the parameter is very small compared to unity. Expansion of Equations (H.2)-(H.7) to first order in yields

(H.10) | ||||||

(H.11) | ||||||

and | (H.12) |

where , , , and

with , and . In the following, for ease of notation, , , and are written simply as , , and , respectively.

According to Table 4.1, the planets in the solar system all possess orbits whose eccentricities, , and inclinations, (in radians), are small compared to unity, but large compared to the ratio of any planetary mass to that of the Sun. It follows that

(H.19) |

which is our fundamental ordering of small quantities. Assuming that , we can perform a secondary expansion in the small parameter . It turns out that when the normalized disturbing function, , is expanded to second order in it takes the general form (see Section H.3)

where is . If we expand the right-hand sides of Equations (H.13)-(H.18) to first order in then we obtain

Note that , , , are , whereas and are .

By analogy, writing the second planet's disturbing function as [see Equation (10.9]

where , and is , and assuming that takes the form

(H.28) |

where is , the Lagrange planetary equations, applied to the second planet, yield

where

(H.35) |