Mathematical model

We can now summarize our mathematical model of the Moon's motion. If $\lambda$ and $\beta$ are the Moon's ecliptic longitude and latitude, respectively, then

$$\cos\lambda\,\cos\beta = \cos\lambda'\,\cos\theta-\sin\lambda'\,\sin\theta\,\cos i,$$ (103)

$$\sin\lambda\,\cos\beta = \cos\lambda'\,\sin\theta + \sin\lambda'\,\cos\theta\,\cos i,$$ (104)

$$\sin\beta = \sin\lambda'\,\sin i,$$ (105)

where

$$\lambda'(~^\circ) = 49.85+13.229\,t+ 6.281\,\cos(13.065\,t-58.3)$$

$$+\,1.678\,\cos(2.078\,t-157.9)+1.228\,\cos(11.318\,t-138.4)$$

$$+\,0.665\,\cos(24.383\,t-109.9),$$ (106)

and

$$i(~^\circ) = 5.146 + 0.1505\,\cos(2.078\,t+118.0),$$ (107)

$$\theta(~^\circ) = 221.8 - 0.05297\,t + 1.678\,\cos(2.078\,t + 22.1).$$ (108)

Here, $t$ is measured in days from 00:00:00 TDT, Jan. 1, 1995.

Figure 53: The residual in the ecliptic longitude of the Moon versus time.

Figure 54: The residual in the ecliptic latitude of the Moon versus time.

Figure 52 shows the ecliptic latitude versus the ecliptic longitude of the Moon for the year 1995, and compares the prediction of our mathematical model against the original data. It can be seen that the two are essentially indistinguishable. Figures 53 and 54 give the residuals in the ecliptic longitude and latitude of the Moon, respectively, versus time, for the years 1995-1996. According to these figures, the residual in the ecliptic longitude generally stays below $10'$, whereas that of the ecliptic latitude generally stays below $1'$. This accuracy is quite sufficient for naked-eye astronomy. Finally, Tab. 3 shows the various orbital elements which we have determined for the Moon, during the course of our analysis, compared to the true values of these quantities. It can be seen that our estimates are remarkably accurate.