Formula to calculate number of sunsets per year based on lattitude How can I create a formula to turn latitude into number of sunsets per year?
Let's keep it relatively simple. Assuming the Earth is a smooth ball, assuming the sun is a single point, not getting involved in different longitudes from year to year, etc.
The formula should return for the North Pole (x = 90) 1 (y = 1). For the Arctic Circle itself (x = 90 - tilt) y = 365 (or 364 if you prefer). All other latitudes in the Temperate Zones should return 365 (or days per year, but let's keep it simple).  The formula should mainly deal with the latitudes in the Polar Regions.
The context of this question is the legal conundrum of how to calculate passing days above the Arctic Circle.
(I'm using sunsets, which is the day change of the Hebrew and other calendars, which is where this question really begins, since other systems might use, let's say, the prime meridian to determine the center of the day, and the day's determination would be straightforward, except for the poles themselves and their immediate surrounding areas.)
If the day only changes when there's an actual sunset then this would create the rather bizarre result of having concentric circles in the Arctic Circle that are wildly off from each other in terms of their calendar date (and this wildness would be exponentially increased if we took into account the differences of longitudes each year, plus altitude, etc., which would create a totally caotic pixelated picture, but let's save that for another time).
 A: I'm sorry, I have this terrible habit of not coming up with answers to my own questions until I ask other people about it.
Here's what I'm thinking. At the shortest day of the year (for the North-Hemishperers), the Arctic Circle itself finally joins his polar latitude friends in not having a sunset. And then, as the Sun creeps North, slowly at first but then picking up speed, until it crosses the North Pole at full speed, the latitudes crossed become "normal" and get sunsets every day. So that sounds to me like a cosine, starting at the Arctic Circle and going North.
(Sorry, stackexchange is being whatever about imbedding pictures)
But that's going from days of year to latitude. The OP wants the other way around. So we can do an arcosine.
Now, I know that this isn't exactly what the OP asked, but being him I happen to know that he will be satisfied with this approach, as well.
So we want x to equal the number of degrees above the Arctic Circle.
So, acos(x) is too short, and not wide enough. (I'm working in degrees, like Earth's coordinate system)
We want x to go up to 23.4, the tilt of the Earth.
So, let's divide x by 23.4.
acos(x / 23.4)
And we have to make it taller so the curve reaches 365.25 (days of the year, about).
Right now the top of the curve is 90 and we want to inflate that to 365.25, so let's multiply our formula by 365.25/90
acos(x / 23.4) * 365.25 / 90
And there you have it.
Now, stackexchange lets me imbed exactly one (1) picture due to my incredibly low reputation, so I'll save that for the end.
Now, to the issue at hand. If each latitude gets a different number of sunsets per year, and the day changes at sunset, so we can drop this formula into excel, using rows as latitudes and columns as years, and color code the cells to get a visual demonstration of what day of the week it is based on your latitude.
Since Excel uses radians, you have to write something like = INT(MOD(DEGREES(ACOS(Latitude/23.4)) * 365.25 / 90 * Years_Passed, 7))

Tada!
Now, I don't actually know if this is correct. Can someone please tell me?
