So far, when considering the distance the sun/moon can be from one of the nodes, we’ve worked out how much the longitudinal and latitudinal parallax impact things and all that’s left now is the fact that the sun and moon aren’t always at their mean position. They both have anomalies which we’ll need to consider. This is because the big goal of this book, so far, is to reduce the amount of math we have to do when checking for an eclipse. While we could go through all the effort of calculating the true position, that’s extra steps. Wouldn’t it be nicer if we could just stop at the mean position if it’s not in the window in which an eclipse can occur?
To that end, our final step in this series of posts exploring the limits for solar eclipses is to translate the true positions to the mean positions.
While I could quote Ptolemy here, the argument he makes is the same as the one he makes when discussing the effect of the anomaly on the lunar limits. In short, it increases it by $3º$.
Thus, if we are to consider the positions of the mean moon and sun, they must be within $11;22º$ of the node when the moon is to the south of the sun at Meroe, or $20;41º$ of the node when the moon is to the north of the sun at the Mouths of the Borysthenes.
That’s it. Much easier than the last few posts.
The only other question we can ask is where this happens, at least as far as in the context of our Table of Mean Syzygies. Specifically, looking at the fifth column, the distance from the Northern limit, the moon will be north of the sun from $0-90$ and then again from $270-360$. Thus, in those contexts we apply the limit for when the moon is north of the sun (i.e., the limit we found for the Mouths of the Borysthenes – $20;41º$). But once the moon crosses the node and would therefore be south of the sun, we apply the other limit (the one we found for Meroe – $11;22º$).
Thus, eclipses in the region defined can only happen when the value in that column are between $69;19º$ and $101;22º$ (near the descending node) or between $258;38º$ and $290;41º$ (near the ascending node). This greatly limits the number of situations we’ll need to check!
