Next: Expansion of planetary disturbing Up: Expansion of orbital evolution Previous: Introduction

# Expansion of Lagrange planetary equations

The first planet's disturbing function can be written in the form [see Equation (10.8)]

 (H.1)

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

 (H.2) (H.3) (H.4) (H.5) (H.6) and (H.7)

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

 (H.13) (H.14) (H.15) (H.16) (H.17) and (H.18)

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)

 (H.20)

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

 (H.21) (H.22) (H.23) (H.24) (H.25) and (H.26)

Note that , , , are , whereas and are .

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

 (H.27)

where , and is , and assuming that takes the form

 (H.28)

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

 (H.29) (H.30) (H.31) (H.32) (H.33) and (H.34)

where

 (H.35)

Next: Expansion of planetary disturbing Up: Expansion of orbital evolution Previous: Introduction
Richard Fitzpatrick 2016-03-31