Lunar Terminator Paradox
notes.secretsauce.net
notes.secretsauce.net
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
Or is this a variant of the "counterintuitive" long shadows of mountains at sunrise/sunset? e.g. https://www.fox13seattle.com/weather/skies-erupt-in-color-du... shadow from underneath the cloud layer.
I'm sure I've seen a picture of the long downwards shadow of a mountain on a desert or plain taken from the top with the sun behind the photographer, but the search seems to be sufficiently AI poisoned now :(
https://en.wikipedia.org/wiki/Crepuscular_rays
A spectacular phenomenon for sure, and I suppose loosely related to lunar phases by virtue of the fact that they both require a very distant source of illumination.
I'm sure there are other places that have similar phenomenon, but Mauna Kea often casts great shadows. Here's one with the moon in the cast shadow on the cloud layer below.
https://twanight.org/gallery/inside-the-shadow-of-mauna-kea/
even though you are very drunk, you don't assume that spinning around will allow you to se more of the basketball than you currently do. even leaning left and right while looking at the ball doesn't give you much of a different view.
the sun is the door. the moon is the ball. you are a person looking at the ball from earth. leaning shifting left a step might give you a perspective that may correlate roughly with moon-at-dusk. shifting right a step might give you a perspective that is roughly moon-at-dawn.
leaning forward or backward, left or right, drunkenly, and spinning around might give you different angles (relative to the line drawn from your ass to your head) that the shiny-side of the basketball seems to be pointing. but you're just drunk. the shiny part of the basketball always points towards the light above the door. it is just your local perspective that is changing.
BASEketball?
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 moon is above the horizon. The sun is below the horizon. Seems pretty straightforward, no?
The trouble is that shouldn't the Earth block the sun's light from reaching the moon then? And the answer is no, because the space is three-dimensional, so the Earth isn't on the straight line(s) connecting the sun and the moon (sometimes it is, but that's what causes lunar eclipses, not the lunar phases).
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.
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.
See this https://chrisjones.id.au/MoonIllusion/ https://astro.unl.edu/naap/lps/animations/moonPhasesHorizonD...
Vsauce: https://youtu.be/Y2gTSjoEExc
"Lunar Terminator Illusion" is the thing to search for. This 11-year old Vsauce video did the trick for me: https://www.youtube.com/watch?v=Y2gTSjoEExc
Check around the 5m3s timestamp if in hurry, and not interested in other stuff he discusses.
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.
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
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.
[1] https://medium.com/@asorlik/the-probability-of-total-solar-e...
https://occultations.org/publications/rasc/2026/lunar26.pdf
I've personally been cataloging those notable stars which are close enough to ecliptic that they have a high probability of occultation, along with the planets that coincide at those times. And it's personally gratifying to watch the Moon occult most or all of the Pleiades, which is also a fairly common occurrence.
The Beehive Cluster is dimmer, and even though the Moon be a waning crescent, for those where it appears above the horizon, may be something exciting to think about!
A better way to describe the geometry is to notice that at sunset when the sun is in the west, if the moon is directly overhead its western half will be illuminated.
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 moment I really _understood_ it... I got a real physical sense of this part of the solar system.
And when you add in the fact that it takes ~5 days (with current human-rated propulsion) to travel from earth to the moon, combined with footage from missions like Artemis II, you can make it even more intuitive.
When I'm out stargazing I also enjoy mentally switching my frame of reference, and instead of thinking about "sunset", imagining being at a certain latitude on the surface of a rocky ball rotating underneath the day/night terminator.
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 ;)