Clouds Passing over the Moon

Our Moon often gets an unfair amount of credit for its luminous properties, considering it acts solely as a pretty average reflector of sunlight.
Earth for instance, all watery and cloudy, reflects sunlight about three times better than the Moon does. You can actually see earthshine being reflected onto the Moon’s surface, and back at us!

Nearly everything we can touch and see is an indirect effect of the Sun being what it is though, so it is a bit trite to diminish the Moon or anything else for being ‘only’ in the Sun’s supporting cast. The Moon certainly is a lot more pleasant to look at, and undoubtedly it has just as important a role in our biological evolution. Not to mention being at the center-point of so much romance.

This morning, the sky was clear and the Moon was free to ‘shine’ — *cough* reflect — the semicircle of sunlight that fell on it from my viewing angle.
I actually flipped the image horizontally because of aesthetics, which means this would be the viewing angle of someone drifting in the ocean south-east of New Zealand, but that’s not very important.

Because the Earth really is round, and so is the Moon and so is the path they both trace around the Sun, these lunar phases are usually more complex in shape than this particular one called ‘First Quarter’.
If you additionally consider their elliptical orbits and shapes, small movements of the Earth, weird things that light does at angles, relativity, …, then things can get very complex. Even things like the lunar phases are not trivial. So we won’t deal with that.

Let’s see what we can say about the simpler case, that of things just passing in front of the moon.
A full moon for instance, getting occluded by a passing cloud.

It’s possible to show how the occlusion of the moon grows and shrinks while the cloud passes over it:

For this particular cloud, if you calculate things out, it appears that during the period the cloud passed over the Moon, we got to see about 27% less moonlight. Meaning, if you have a thing which is sensitive to it — for instance, if you happen to be a werewolf, your anger and bloodlust could be such a thing— , it would have gotten 27% less effect than without the cloud.

This 27% is related to the shape of the cloud. Not only to its own area but in the way it can “use” that area to prolong its occlusion of the Moon for as long as possible.

Let’s try to consider clouds that at some point in their passage completely occlude the moon, but both start and end with a full moon, and take exactly “one moon width” to pass over the Moon.
The easiest example is a cloud that has the exact same shape as the moon in the sky, and passes over it.

There is one perhaps non-intuitive thing that happens for the situation when this circular cloud passes over the Moon — and for many other types of clouds actually. It has to do with the way the shaded area increases and decreases in this case:

You will notice at the top there is a “spike”, it isn’t smooth!

The reason for this is that the shaded area grows faster as it starts occluding the Moon.
As the cloud edges closer to covering the entire moon, it gets more and more bits of circle real-estate to ‘eat’ in the form of a larger but thinner crescent, and it grows in size at a faster rate. Of course, when it slips away out of the Moon again, the situation is symmetrical and it starts diminishing quite quickly before slowing down.
From a mathematical perspective, this means the top point is non-differentiable. Depending on how you’re wired, this may seem non-intuitive — it did to me!
Perceptively you might have felt this happen when you cover a circle with another circle or have witnessed a lunar eclipse: it suddenly “clicks” with no feeling of wiggle-room and it is easy to tell whether it is covering the entire circle or whether it is just not quite right.

You can contrast this with the way a square cloud would pass over the moon:

In this case, something different happens: during the first half of the moon (which stops at 1/4th of the above sequence), the rate at which it covers it is increasing. But, as it then goes on to cover the entire moon, there gets less and less circle to “eat”, so the rate slows down.
From a mathematical perspective this means, for the square cloud, the top is indeed differentiable.
Perceptively, you will get a more gradual occlusion, and it will be harder to guess if it has really covered it 100%, because there’s just a tiny sliver of moon to be seen to the right or left, instead of the crescent with the circular case, which is easier to use as a visual cue.
Here you can see how the area changes over time, and the yellow curve to tell you how much it varies at each step. Note these are not stretched out sine waves, they’re rather related to the derivative of arccos(x), which has a similar shape but is not quite the same.

How do these two clouds compare? Well, if you calculate everything, which for the circle involves integrating over the formula you can get for A here, you end up with the following:

The square cloud will cause us to only see 50% of moonlight during its passage, and the circle cloud will cause about 57.6% of moonlight. Actually the accurate outcome for the latter will be 100/(π/(π - 4/3)) once you write it all out.
So, the square cloud, as expected, actually occludes the most sunlight.

However, there is still a family of cloud shapes that outperforms the square. Can you guess which?

🥁

Here it is:

This contrived cloud shape creates the largest amount of occlusion of the moon.

It only leaves 42.4% of moonlight to reach us (you’ll notice the symmetry with the circle case which falls at the other end of 50%).
If you add the differential information you can see that it immediately starts eating the moon surface, at a rapid pace, slowing down in the middle before picking up the pace again. There isn’t any way to make a more efficient cloud.

There are of course infinite variations of these clouds, so you can make them look a little more “real”, but these theoretical ones set the boundaries: the circle as the most “moon-friendly” case, and the strange cloud as the least. Everything else falls in between these!

Things can get pretty complex if you consider objects that do not only slide, such as clouds, but also rotate. You might have, for instance, a satellite passing in front of the moon, or a boy on a bicycle with spinning wheels.
Or, this particular spaceship spinning across the moon:

Calculating the actual area in this case could take a very long time to get right. Even if you started you’d probably give up halfway.
One way to deal with this situation is to just draw a high-resolution image and count the pixels of the overlap area.
This is a rather funny way of replacing deep calculations with a cop-out. While you can use voxels in a similar way if you want to count a 3D volume, there are considerable limitations with those that you will not face in the 2D case, which is something I thought interesting to mention.

If the ‘moon’ is no longer circle-shaped, things can get extremely complex to calculate, such as this case of occlusion, which by the way is 25%.

One interesting thing to experiment with is to think of a shape of the area curve (for instance a parabola, or a line going up), and see if you can invent an object that, when it passes over a circular moon in a certain way, creates that type of curve. Integral calculus can help you get there, but intuition will probably end up being faster.

Now, the thought I had has also passed, and I hope it triggered your brain somewhat, even if it’s only mildly interesting.
Concerning the topic of occultation, which is a great word to use to stop conversations, there’s a pretty neat telescope design that abuses the blocking of starlight to see more, you can read about it here.

One last advice, don’t forget dreaming about the Moon, it’s good for you!