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)
$\displaystyle b_8$ $\displaystyle = 0.$ (11.232)

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

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

According to Equations (11.136) and (11.148),

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

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

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

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

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

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

$\displaystyle x_{10}$ $\displaystyle = \frac{1}{2},$ (11.241)
$\displaystyle y_{10}$ $\displaystyle =-\frac{11}{16}.$ (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)
$\displaystyle \delta \beta$ $\displaystyle = 0.$ (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-TouzĂ© and Chapront 1988)] because of the omission of higher-order (in $m$ and $e'$) contributions.