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I am reading Longest Sunset from XKCD. At first, I've thought that he mentions these phenomena just for fun, but it turns out to me that it might happens, based on the level of somewhat seriousness that I perceive from his writing. He doesn't talk about that in detail.

For the sunset:

enter image description here

Sunset starts the instant the Sun touches the horizon, and ends when it disappears completely. If the Sun touches the horizon and then lifts back up, the sunset is disqualified.

I'm not sure if the sunset is disqualified is because it happens, but we don't count it, or because he's just making fun. But I can't prove that this cannot happen. The more we head to the poles, the shorter the night is. Ultimately there will be a point that the sun still sets, but not completely, right?

For the sunrise:

For the purposes of our question, this is not a sunset:

enter image description here

The phrase for our purposes strengthen my doubt that he is being serious. The logic is the same above.

But these two illustrations are in the series of other apparently amusing ones, which are the sun as the cell in division, or as the egg in hatching (if you read the book, you will see this effect stronger).

So, is there an instance where the sun sets or rises partly, then return back?

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Yes, such sunrises happens every year at the beginning and end of the polar nights at high latitudes. One can have a few days with a glimpse of the sun but disqualified sunsets and sunrises.

Sunsets occur at the end of the midnight sun period by the end of the summer, the first sunset is not complete.

My own picture from Svalbard. Firs sunrise at Tempelfjorden. Sun at it's highest elevation at midday. (To be honest, the picture might be taken the day after the first sunrise, as some mountains are hiding the horizon.)

Here is e.g. an article from Svalbardsposten, the northernmost newspaper in the world, reporting of the first rays of sunlight after the polar night and some pictures from the last sunrise/sunset before the polar night in northern Sweden.

The same occurs at the southern polar circle and south thereof, unless it's cloudy...


Update: here is a great time laps from Davis Station in the Vestfold Hills showing exactly what you asked about: Mid winter

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