Lunar Terminator Paradox
notes.secretsauce.net101 points by dima55 a day ago
101 points by dima55 a day ago
To me, this article conflates lunar elevation and phase. Together with the use of ambiguous terms "up", "down", "above", and "below" makes it difficult to understand the author's intent and what their software is meant to visualise.
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
The strange thing is that there's only visualizations and no photographs! It's not like it's difficult to find the moon to take photos of, nor the sunset.
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 :(
I think you're looking for crepuscular rays.
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.
Beautiful picture of Mt Rainier!
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/
The only place where the illuminated part of the moon points straight down, as the article expects, is also the only place where the sun moves perpendicular to the horizon when setting: if you're at the equator.
Doesn't the sun do this at all latitudes at the equinoxes? (Rises due east, sets due west)
In that time lapse, when the moon is visible, the illuminated portion is always facing “down,” i.e. the horizon?
You walk into a dark basketball court through a doorway with a single bright light above it. as you approach the free-throw line, you realize someone has left a basketball at center court. part of it is lit by the light of the doorway.
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.
Is that the Alien version of the Voight-Kampff test?
More like the Moneyball rational fiction with robots version of it. Or whichever movie was about basketball.
> Now, what happens if the moon is not at the horizon, but higher up at the sky?
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.
> Also it's hard to reason about up and down
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).
Was this known as the "Lunar Terminator Paradox" or called that anywhere before this article? So far as I can tell this seems to be entirely made up and the only references I can find are to TFA or related.
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.
There’s no paradox present, just one person’s misunderstanding of the arrangement of celestial bodies in 3D space.
The sky is spherical, so straight lines in the sky aren’t straight lines on its 2D projection on the retina. Rather, straight lines are great circles. So if you trace the great circle that connects the sun and the moon, the moon should "point" along that great circle – which can be considerably off of what we think of as a straight line (which is curved in reality).
Well sure. Yeah. Do you find this explanation clear and satisfying though? Can you use it to answer a question like "at sunset, during a half-moon, can you see an upward-facing moon"? Maybe you're much smarter than me, but I had to write the simulator to figure out the answer. Claude and gemini never wrote the simulator, so they consistently answer that question incorrectly :)
No, you will not see an "upward facing" moon. At sunset, the west side of a half the moon will be illuminated. I think you are getting your self tangled with terms like "upward" and "downward" and "facing"
Makes perfect sense to me. When starting the shortest path for a flight from San Francisco to Tokyo, you don't point the plane toward Japan, you point it towards Alaska.
Well, you also point it at Japan.
Indeed, ideally you'd point it at the only path that is pointing straight towards Japan. No other path actually points towards Japan (except the one going in the opposite direction, but that's longer).
This isn’t really a paradox if you just visualize the Moon as a sphere and visualize where the Sun is shining from, and I think that’s what the plots in OP demonstrate.
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.
I understand less than before I read this article.
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?
> And what on earth is the first plot? A moon that rises to its zenith doesn't become half lit
If it rose to the zenith, it would. This is the paradox.
Huh ? You do realise that the apparent motion of the moon over a given day is because of the rotation of the earth, and that the changes in the relative positions of the sun and moon, and hence the lighting conditions of the moon, are negligible ?
But then it wouldn't be sunset anymore. I understood the whole article to be about things you can see at sunset.
But a full moon can't rise to it's zenith at sunset.
But if it did, then it would be a half moon.
Of course not, because the sun is much further away than the moon.
If the moon is directly above you, and the sun is directly in front of you, they are at right angles from you.
This article is wrong at so many levels
See this https://chrisjones.id.au/MoonIllusion/ https://astro.unl.edu/naap/lps/animations/moonPhasesHorizonD...
Vsauce: https://youtu.be/Y2gTSjoEExc
A DIY explainer with photos https://www.metabunk.org/threads/the-moon-tilt-terminator-il...
This article was quite hard to understand, and the author called it "Lunar Terminator Paradox", which does not give too many Google hits.
"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.
Simple way to imagine this in your head.
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 don't seem to be understanding the paradox. The lit side of the moon is the top side at sunset when the sun is behind you and the moon is in front of you? But if the moon and the sun were in the same direction it would be the bottom side. It's still facing the sun, it just looks opposite to you because you turned yourself around to face the other way.
is this a bad moment to mention how weird it is that the moon is perfectly positioned to create eclipses?
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
The moon is 400 times smaller than the sun, but the sun is 400 times farther away, so they appear to be the same size. The solar eclipse we see, with the moon obscuring the sun but just allowing light around the edges the way it does, may be a very uncommon phenomena.
it was my vague understanding that what is weird is the apparent size matching. since the moon is slowly moving away, eventually we'll no longer have total solar eclipses, just annular and partial.
The relative sizes are approximate, not perfect, but close enough that our monkey brains interpret it as more than a coincidence.
Idk, one could come up with a hypothesis that eclipses provide the first tangible validation of a system of mathematics.
If there were greater significance to it, I’d have thought the the visual sizes of the Sun and Moon would have been perfectly matched, rather than to within a few percent, give-or-take.
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.
<distant growling low resonating voice that makes galaxies shake> Damn, those pesky IEEE-754 accumulation errors again.
It is never a bad moment to mention this. For planets with a single moon, the probability of such an occurrence has been estimated at 4% [1]. I like to think that this is what extraterrestrials would find most fascinating about our world.
[1] https://medium.com/@asorlik/the-probability-of-total-solar-e...
It’s also transient. In a few million years, the Moon will have drifted far enough away that it cannot fully eclipse the sun. Long ago, the Moon was so close you wouldn’t have been able to clearly see the Sun’s corona during a total eclipse.
Over 600 million years left! Complex life may be gone by then (see silicate weathering).
It’s more like, if the moon were much farther away then total solar eclipses wouldn’t happen at all. But the opposite is not true. If it were closer than total solar eclipses would be more common.
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.”
Im not trying to be mean but nothing you said made any sense, possibly because I wasnt clear.
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.
By the way, current Stellarium+ simulations indicate that the Moon is positioned to plow through occultations of the Beehive Cluster and Jupiter, albeit below the horizon for me, coming up in the first week of October. There will also be a very close encounter with Mars... which happens every month, anyway.
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!
I think it's easier to understand if you consider the opposite case: how could the moon never be on the Sun-Earth ray? It would have to orbit Earth around the Sun-Earth ray. But the Sun-Earth ray changes direction as Earth orbits the sun, so the Moon's orbit would have to change with it. People in the northern hemisphere would not be able to see the moon at the same time as people in the southern hemisphere for significant portions of the Lunar cycle, and we would always see a half-moon.
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.
I was also confused by the article, but I've noticed that I'm not the only one. I think I get what confused the author, it's the fact that angle the shadow appear to us depends on the latitude of the viewer on Earth. It's not super intuitive.
The sun doesn’t set because the sun/earth moves, it sets because the earth spins. There’s no reason the setting sun should change which part of the moon is illuminated.
Reminds me of debugging a visual glitch that was just a bad assumption about perspective. Classic 'it's not a bug' scenario.
Had to write a shader once that implicitly dealt with this. Always wild how the terminator appears to speed up near the poles.
Perhaps the render should use the real sizes and positions in meters for all three objects, at a time given by the user
Author uses the words "top" and "bottom" but these words are ambiguous when it comes to objects in the sky.
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.
Was expecting time travelling moon robots. :(
Omg this paradox has bothered me for so long. I’m thrilled to have it finally explained.
The TLDR is that the sun is very, very far away compared to the moon.
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.
^ THIS!
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.
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It’s cool and all, but I do worry about this instinctive reaching for AI to build convoluted answers to stuff like this, which is easily demonstrated by, say, two balls and a lamp, not to mention the myriad of astronomical simulators that already exist.
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...
I think you're overstating your case. I never said this was novel. I didn't vibe code any of this. Two balls and a lamp were not sufficient to make this clear to me, but writing the simulator was.
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.
> is easily demonstrated by, say, two balls and a lamp,
...have done this many times for friends! A torch and a yoga ball work nicely ;)