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# Expansion of planetary disturbing functions

Equations (10.8), (10.9), (H.1), and (H.27) give

 (H.36) and (H.37)

where

 (H.38)

and

 (H.39) (H.40)

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 Laplace coefficients. Thus,

 (H.53)

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),

 (H.64)

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

 (H.69)

where

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

 (H.71)

and

 (H.72)

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

 (H.73) (H.74) and (H.75)

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

 (H.76) (H.77) and (H.78)

Next: Derivation of Gauss planetary Up: Expansion of orbital evolution Previous: Expansion of Lagrange planetary
Richard Fitzpatrick 2016-03-31