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Next: Parallactic inequality Up: Lunar motion Previous: Variation

Annual equation

Next, let us consider terms in the solution of the lunar equations of motions that depend linearly on the solar eccentricity, $ e'$ .

According to Equations (11.135) and (11.147),

    $\displaystyle a_8$ $\displaystyle = \frac{3}{2},$ (11.231)
and   $\displaystyle b_8$ $\displaystyle = 0.$     (11.232)

It follows from Equations (11.192), (11.193), and (11.201) that

    $\displaystyle x_8$ $\displaystyle = \frac{3}{2}\,m,$ (11.233)
and   $\displaystyle y_8$ $\displaystyle =-3.$ (11.234)

According to Equations (11.136) and (11.148),

    $\displaystyle a_9$ $\displaystyle = \frac{21}{4},$ (11.235)
and   $\displaystyle b_9$ $\displaystyle =-\frac{21}{4}.$     (11.236)

It follows from Equations (11.192), (11.193), and (11.202) that

    $\displaystyle x_9$ $\displaystyle = \frac{7}{2},$ (11.237)
and   $\displaystyle y_9$ $\displaystyle =-\frac{77}{16}.$ (11.238)

Finally, according to Equations (11.137) and (11.149),

    $\displaystyle a_{10}$ $\displaystyle = -\frac{3}{4},$ (11.239)
and   $\displaystyle b_{10}$ $\displaystyle =\frac{3}{4}.$     (11.240)

Equations (11.192), (11.193), and (11.203) yield

    $\displaystyle x_{10}$ $\displaystyle = \frac{1}{2},$ (11.241)
and   $\displaystyle y_{10}$ $\displaystyle =-\frac{11}{16}.$ (11.242)

It follows from Equations (11.122)-(11.124), (11.167)-(11.169), and (11.179)-(11.181), as well as the previous expressions for $ x_8$ , $ x_9$ , $ x_{10}$ , $ y_8$ , $ y_9$ , and $ y_{10}$ , that the net perturbation of the lunar orbit due to terms in the solution of the lunar equations of motion that depend linearly on $ e'$ is

    $\displaystyle \delta R$ $\displaystyle =\frac{3}{2}\,m^{\,2}\,e'\,\cos{\cal M}' -\frac{7}{2}\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')+\frac{1}{2}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}'),$ (11.243)
    $\displaystyle \delta \lambda$ $\displaystyle =-3\,m\,e'\,\sin{\cal M}'+\frac{77}{16}\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}') -\frac{11}{16}\,m^{\,2}\,e'\,\sin(2\,D + {\cal M}'),$ (11.244)
and   $\displaystyle \delta \beta$ $\displaystyle = 0.$     (11.245)

The previous expression are accurate to $ {\cal O}(m^{\,2}\,e')$ .

The first term on the right-hand side of Equation (11.244) is known as the annual equation, and is caused by a combination of the perturbing action of the Sun and the slight eccentricity ( $ e'=0.016711$ ) of the apparent solar orbit about the Earth-Moon barycenter. The annual equation attains its maximum amplitude when the Earth (or, rather, the Earth-Moon barycenter) is halfway between its perihelion and its aphelion points (i.e., when $ {\cal M}'=90^\circ$ or $ 270^\circ$ ). Conversely, the amplitude of the annual equation is zero when the Earth passes through its perihelion or its aphelion points (i.e., when $ {\cal M}'=0^\circ$ or $ 180^\circ$ ). According to Equation (11.244), the annual equation generates a perturbation in the lunar ecliptic longitude that oscillates with a period of a solar year, and has an amplitude (calculated using $ e'=0.016711$ and $ m=0.07480$ ) of $ 773$ arc seconds. As before, the oscillation period is in good agreement with observations, whereas the amplitude is somewhat inaccurate [it should be $ 666$ arc seconds (Chapront-Touzé and Chapront 1988)] because of the omission of higher-order (in $ m$ and $ e'$ ) contributions.


next up previous
Next: Parallactic inequality Up: Lunar motion Previous: Variation
Richard Fitzpatrick 2016-03-31