next up previous
Next: Summary of results Up: Lunar motion Previous: Evection in latitude

Reduction to ecliptic

Finally, let us calculate the remaining terms in the solution of the lunar equations of motion.

According to Equation (11.153),

$\displaystyle c_2 = \frac{3}{2}\,x_1\,z_1.$ (11.321)

It follows from Equations (11.209), (11.211), (11.290), and (11.315) that

$\displaystyle z_2 = -\frac{3}{2}$ (11.322)

According to Equation (11.154),

$\displaystyle c_3 = \frac{3}{2}\,x_1\,z_1.$ (11.323)

It follows from Equations (11.209), (11.212), (11.290), and (11.315) that

$\displaystyle z_3 = \frac{1}{2}$ (11.324)

According to Equation (11.127),

$\displaystyle a_{03} = \frac{3}{4}\,z_1^{\,2}=\frac{3}{4},$ (11.325)

where use has been made of Equation (11.315). Equation (11.189) yields

$\displaystyle x_{03} =-\frac{1}{4}.$ (11.326)

Finally, according to Equations (11.130) and (11.142),

    $\displaystyle a_3$ $\displaystyle =-\frac{3}{4}\,z_1 = -\frac{3}{4},$ (11.327)
and   $\displaystyle b_3$ $\displaystyle = 0,$     (11.328)

where use has been made of Equation (11.315). Hence, Equations (11.192), (11.193), and (11.196) yield

    $\displaystyle x_3$ $\displaystyle =\frac{1}{4},$ (11.329)
and   $\displaystyle y_3$ $\displaystyle =-\frac{1}{4}.$ (11.330)

It follows from Equations (11.122)-(11.124), (11.159), (11.162), (11.174), (11.185), and (11.186), as well as the previous expressions for $ x_{03}$ , $ x_3$ , $ y_3$ , $ z_2$ , and $ z_3$ , that the net perturbation to the lunar orbit due to the remaining terms in the solution of the lunar equations of motion is

    $\displaystyle \delta R$ $\displaystyle =0$ (11.331)
    $\displaystyle \delta \lambda$ $\displaystyle =-\frac{1}{4}\,I^{\,2}\,\sin(2\,F),$ (11.332)
and   $\displaystyle \delta \beta$ $\displaystyle =-e\,I\,\sin(F-{\cal M}) + e\,I\,\sin(F+{\cal M}).$     (11.333)

These expressions are accurate to $ {\cal O}(I^{\,2})$ and $ {\cal O}(e\,I)$ .

All of the terms on the right-hand sides of Equations (11.332) and (11.333) are Keplerian in origin (i.e., they are independent of the perturbing influence of the Sun). The term on the right-hand side of Equation (11.332) is due to the slight inclination of the lunar orbit to the ecliptic plane, and is known as the reduction to the ecliptic.


next up previous
Next: Summary of results Up: Lunar motion Previous: Evection in latitude
Richard Fitzpatrick 2016-03-31