## Are shadows cast by moonlight larger than those by sunlight?

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systemchalk
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### Are shadows cast by moonlight larger than those by sunlight?

I should begin by saying this question is inspired by homework, though it was philosophy and the answer is not required to complete the assignment. For those curious (and by way of a set up) the question described a man from ancient greece setting up two torches at different distances from each other and using the fact that the closer torch appears brighter as evidence that the sun is closer to the earth than the moon. The student has to come up with an argument intelligible to the speaker (excepting, of course, that we are not expected to write the answer in ancient greek).

I originally thought to point out that a shadow grows larger the closer an object gets to a light source, and so a simple experiment could be set up to measure the distance of a shadow caused by sunlight and another caused by moonlight (obviously at the same angle). The difficulty is that while my intuition says this should be the case, I really have no way of knowing if this is what actually happens and nobody has seen fit to put this vital information on the internet.

The xkcd forums seemed to be the most logical place to ask such a question and finally get the nagging question out of my head.

Hero Polymer
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### Re: Are shadows cast by moonlight larger than those by sunli

The short answer: I think yes.

The unhappy answer: I think no, at least not enough to be detectable by any experiment possible by Ancient Greeks.

However... You could probably set up a demonstration using the same torches.

elasto
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### Re: Are shadows cast by moonlight larger than those by sunli

They wouldn't be any different because the sun and moon appear approximately the same size in the sky, so the rays of light from the edges of each will be in the same direction.

Since it's a homework assignment I won't spoonfeed it, but I will say if you Google 'measuring the distance to the sun' you'll find they managed to do it even in Ancient Greek times and got answers that were pretty accurate

And, for sure, they knew the sun was much further away than the moon - even the lay people would probably have known it due to a once-in-a-lifetime or so astronomical event visible to all...

Technical Ben
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### Re: Are shadows cast by moonlight larger than those by sunli

Yep. always remember eclipses when thinking about the moon and sun. Why? It's important they happen because they both have the same apparent size when viewed from earth.
Solves quick questions like "how much bigger is the sun in the sky than the moon"
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Gigano
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### Re: Are shadows cast by moonlight larger than those by sunli

Technical Ben wrote:Solves quick questions like "how much bigger is the sun in the sky than the moon"

You'd need to know their distances to the earth though to figure that out, right?
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Timsk
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### Re: Are shadows cast by moonlight larger than those by sunli

To quote Simon Singh (Big Bang): [cite]the Greeks had shown how knowing the diameter of the sun depends on knowing the distance to the sun, which depends on knowing the distance to the moon, which depends on knowing the diameter of the moon, which depends on knowing the diameter of the Earth.[/cite]

I'm proud that my first post here is a quote from that book!

Oh and... Hi!

Technical Ben
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### Re: Are shadows cast by moonlight larger than those by sunli

The rider to the question was "in the sky", I meant "apparent size visible from earth", not "actual size". So the answer is "the same (minus a tiny bit for the moon AFAIK)".
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AvatarIII
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### Re: Are shadows cast by moonlight larger than those by sunli

iirc the Sun's diameter is about 400 times that of the Moon, and about 400 times further away.

the simplest solution would be the eclipse thing. Assuming the Greek knows that eclipses are caused by the moon, the fact that the moon moves in front of the sun proves that the moon is closer.

speising
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### Re: Are shadows cast by moonlight larger than those by sunli

I would first point out that the greek has to prove first that sun and moon have he same absolute luminosity. Which is of course not the case.

gmalivuk
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### Re: Are shadows cast by moonlight larger than those by sunli

Technical Ben wrote:The rider to the question was "in the sky", I meant "apparent size visible from earth", not "actual size". So the answer is "the same (minus a tiny bit for the moon AFAIK)".
Sometimes minus a bit, sometimes plus a bit. Remember that there are also total eclipses on occasion.
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### Re: Are shadows cast by moonlight larger than those by sunli

AvatarIII wrote:iirc the Sun's diameter is about 400 times that of the Moon, and about 400 times further away.

Nope. Distance to the sun is about 150 million km. Distance to the moon about 144 thousand km (alternatively: 8 light minutes and 1/2 a light second). So a factor 1000 different.

the simplest solution would be the eclipse thing. Assuming the Greek knows that eclipses are caused by the moon, the fact that the moon moves in front of the sun proves that the moon is closer.

Of course they knew this. You can see the moon move in front of the sun, it's not like the sun just disappears out of nowhere.
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phlip
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### Re: Are shadows cast by moonlight larger than those by sunli

I know we've started to go off on a tangent here, but I figured I'd answer the original question anyway.

For ease of standardisation, we'll say that you're suspending a 1m-wide object, 1m above the ground, and seeing what size shadow it casts. For instance, if you had a point light source 3m above the ground, directly above our object, it would cast a 1.5m-wide shadow. Also, for simplicity, we'll say the sun or moon is directly overhead (so you're doing this at noon or midnight, at the appropriate latitude. And it's a perfect full moon, without being a lunar eclipse somehow... just go with me on this.)

If you treat the sun and moon as point-sources of light at their respective distances, then, as you expect, the shadow from the moon is larger. How much larger? Well, the shadow from the sun is about 1m plus 6.6 to 6.8pm wide (depending on where we are in our orbit, as we get closer and further from the sun), while the shadow from the moon is about 1m plus 2.5 to 2.8nm wide. So we're talking a difference of a couple of nanometres. Now, to even resolve a difference that small, the light you're using has to have a wavelength on at least that order of magnitude... so we're talking EUV/low X-Rays and higher... visible light simply can't resolve a difference that small... we're talking differences where the light diffracting off the corner of the object is enough of a problem. Alpha has a few other comparisons for the size... like approximately the radius of a DNA helix, or 1/5 to 2/5 of the thickness of a cell membrane. So yeah, unless these are Ancient Greeks who can detect DNA-sized changes in xray shadows...

However, the main complication comes from the fact that they aren't point sources of light, and are actually quite big. This means that the shadows they cast are going to be quite fuzzy around the edges, as the point that previously was considered to be the "edge" of the shadow is now merely the point at which our object covers half the sun/moon... so it'll be 50% in shadow. And for some distance on either side, it will fade to 0% or 100%. This is where the fact that both the sun and moon have roughly the same angular size in the sky comes into play, as it means that this fuzziness is about the same for both the sun and the moon... it also varies depending on how far we are from the sun/moon, but it's going to be around 8.5-9.5mm wide. Millimetres. That's going to totally swamp the nanometre-scale changes we're looking to measure... and you need to somehow find the exact middle of this fuzzy area to get the measurements we were expecting in the previous paragraph.

So... technically the difference you're looking for is there, but completely impractical to measure.

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speising
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### Re: Are shadows cast by moonlight larger than those by sunli

AvatarIII wrote:iirc the Sun's diameter is about 400 times that of the Moon, and about 400 times further away.

Nope. Distance to the sun is about 150 million km. Distance to the moon about 144 thousand km (alternatively: 8 light minutes and 1/2 a light second). So a factor 1000 different.

Actually, the distance to the moon is about 380000km.

Technical Ben
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### Re: Are shadows cast by moonlight larger than those by sunli

speising wrote:
AvatarIII wrote:iirc the Sun's diameter is about 400 times that of the Moon, and about 400 times further away.

Nope. Distance to the sun is about 150 million km. Distance to the moon about 144 thousand km (alternatively: 8 light minutes and 1/2 a light second). So a factor 1000 different.

Actually, the distance to the moon is about 380000km.

I think I'm confused. Let's let our good old friend citation fix that.

Semi-major axis 384,399 km

http://en.wikipedia.org/wiki/Moon

Speising wins the round!

The moon is about 1.3 Light Seconds away.

The incorrect number at 144 thousand km is 0.4 light seconds not the 8 Light minutes. This sounds like the time light travels from the Sun to moon? But it's not 144000 km, no idea where that came from though... http://www.wolframalpha.com/input/?i=144000km
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AvatarIII
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### Re: Are shadows cast by moonlight larger than those by sunli

AvatarIII wrote:iirc the Sun's diameter is about 400 times that of the Moon, and about 400 times further away.

Nope. Distance to the sun is about 150 million km. Distance to the moon about 144 thousand km (alternatively: 8 light minutes and 1/2 a light second). So a factor 1000 different.

I guess I remember incorrectly then, not sure where I got the 400 figure from.

edit:
according to wikipedia
The average distance from the Earth to the Moon is about 1.282 light-seconds
The average distance from the Earth to the Sun is 499.0 light-seconds.
499 / 1.283 = 389.2

diameter of the moon = 3 474.8 kilometres
diameter of the sun = 1 391 000 kilometres
1,391,000 / 3,474.8 = 400.3

I guess I got the 400 figure FROM REALITY!

elasto
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### Re: Are shadows cast by moonlight larger than those by sunli

Ok, so a couple of days has passed so it's probably ok to 'spoil'. This is a pretty good link on how an Ancient Greek found approximate distances to the moon and sun: http://en.wikipedia.org/wiki/Aristarchu ... _Distances

gmalivuk
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### Re: Are shadows cast by moonlight larger than those by sunli

Technical Ben wrote:The incorrect number at 144 thousand km is 0.4 light seconds not the 8 Light minutes. This sounds like the time light travels from the Sun to moon? But it's not 144000 km, no idea where that came from though...
No one ever said 144,000km was 8 light minutes. It was described instead as half a light-second, which it is.
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