Hold two ping pong balls at arms' length, one above and further away than the other. Right now you are the sun.
Put a red dot on top of the lower one, and then rotate it just slightly until you can't see it. This is the observer on Earth, who is after sunset.
Put a red dot directly in the middle of what you can see on the upper ball. This is the center of where the sun is striking the moon.
Keep looking at this dot on the moon. Now, keeping their relative positions fixed, bring the balls towards you and up, until you are looking from the perspective of the observer on Earth. What happens to the dot on the moon? It appears to rise up away from you.
That's the paradox.
Lunar elevation (aka "altitude" in celestial coordinates) is sidereal, so the maximum range is determined by the viewer's latitude. The moon's orbital plane is not perfectly aligned with that of the sun-earth so there's another component there, but this only varies +/- 5 degrees of the sun's declination.
Lunar phase is synodic, so there's no significant variation in the observed illuminated portion with the moon's elevation over one night.
The full moon occurs when the moon is on the opposite side of the earth from the sun, so it rises at sunset, transits the meridian at midnight and sets at sunrise, (+/- a few minutes for seasonal variation).
Here's a nice visualisation of the lunar analemna - the figure described by the apparent position of the moon over a month - from Mt. Laguna at 32.8 degrees north. https://www.hpwren.ucsd.edu/news/20250212/
I also found this page with a decent explanation of the apparent effects of orbital cycles. https://www.cyclecalcs.com/learn/synodic-sidereal.html#faq
Moon follows the sun to the west so to move moon "higher" up in the sky you need to rewind time (15deg/hour). In e.g. 4 hours moon barely moves in it's 28 day cycle so the moon will look exactly the same (you can take a picture and compare). It's the earth rotation that make moon "go down" or up. I've read your article 2 times and I still don't understand what is apparent problem, no wonder AI had trouble. Also it's hard to reason about up and down when moon move along ecliptic which is curve. Down at noon points elsewhere as down at sunset.
The “paradox” might be in the initial assertion which does not use an accurate mental model: “After sunset, the sun is below, so we would expect the illuminated portion of the moon to point down, towards where the sun is.” This assertion might be true if at sunset the Sun ducks just behind the horizon, around the same distance or closer than the Moon is to the Earth. In reality, the Sun is of course much farther from Earth than the Moon. Factoring the relative distances into the mental model will lead you to the correct conclusion.
As others have noted, there doesn't even seem to be an actual paradox other than a possible confusion about how a half-lit sphere looks from various perspectives.
When has anyone ever noticed a full moon becoming less full during the night? Not that the effect doesn't happen, but it is minor. This effect also occurs during first and last quarter, despite what the article claims.
And what on earth is the first plot? A moon that rises to its zenith doesn't become half lit. And is the dark coloured part li?, Where is the 3/4 moon?
If it rose to the zenith, it would. This is the paradox.
I think its pretty incomprehensible odds if you ask google.
But I could be wrong I know literally nothing about astronomy, it just always surprised me its not common, but it be cool if someone could explain that
[1] https://medium.com/@asorlik/the-probability-of-total-solar-e...
Also since the Sun-Earth distance varies and so does the Earth-Moon distance, the coverage of the Sun varies. We sometimes get annular eclipses where the Sun is not totally covered.
It is an interesting coincidence. But the statistician in me feels like calculating the probability is a bit spurious. For one, not all combinations of distance and size are equally likely, and running physical simulations to try and get decent Monte Carlo estimates of the odds feels a bit suspect - I imagine a decent chunk of what you’d be measuring is the influence of parameters whose values are empirically unknowable.
But, even more than that, it comes from the same place that requires me, when a cashier sees the total is exactly $50 and asks, “What are the chances of that?” to actively suppress the urge to say, “About the same as for $59.37.”
I do appreciate your time and your feedback and Im not great at patience but Im going to try because I am dim in many subjects.
I meant an exact solar eclipse where the Sun and the moon seem to match perfectly.
As such the math odds are low.
Finally, if I ever went to a shop and I was charged a round number like $50, or in my case even $3 I would immediately assume I am being scammed, as that is rare given the context. ( which incidentally happened to me once in Barcelona due to a local spanish barman trying to charge me a tourist tax ).
But I am grumpy and it is late, please forgive coarseness of my reply.
That the Earth-Moon orbital axis is not incidental with the Sun-Earth orbital ray means we'd always, eventually, have solar and lunar eclipses.
Perhaps if Earth were in a tidally locked orbit with the Sun, the Moon could be in a tidally locked orbit with the Sun while also orbiting the Earth. But then half the planet would be frozen and the other half would be completely baked and we wouldn't be having this conversation.
It’s actually quite fun to look at the moon, adjust your mental model based on where the sun is relative to you and what you see illuminated, and experience the scale of the solar system.
The software is being written to teach the user something, but it’s not novel software, or a novel problem, and the creator doesn’t actually care about the process of making the software, or actually thinking about how to solve the question. But they have access to an automatic wheel reinvention machine and so...
The LLMs have read all of the internet, and are thus quite good at answering questions like this, and I have zero problems asking them about it. In THIS case, the answers available on the internet kinda suck, which made the LLM suck as well.
...have done this many times for friends! A torch and a yoga ball work nicely ;)