What is the curve on the inside of a cresent moon?

[halfeye]

It's not an arc of the same radius as the outer, is it circular at all?

[tiornys]

Not an astronomer, but it seems like it should be the arc of the earth's circular shadow where it intercepts the moon. However, a quick spot of research suggests otherwise. From Wikipedia's entry on "Cresent":

The shape of the lit side of a spherical body (most notably the Moon) that appears to be less than half illuminated by the Sun as seen by the viewer appears in a different shape from what is generally termed a crescent in planar geometry: Assuming the terminator lies on a great circle, the crescent Moon will actually appear as the figure bounded by a half-ellipse and a half-circle, with the major axis of the ellipse coinciding with a diameter of the semicircle.

So that inside curve is apparently half of an ellipse.

[halfeye]

Not an astronomer, but it seems like it should be the arc of the earth's circular shadow where it intercepts the moon. However, a quick spot of research suggests otherwise. From Wikipedia's entry on "Cresent":



So that inside curve is apparently half of an ellipse.

No, that's not even close to right, half an ellipse would mean that the arcs near the poles were narrowing much more rapidly.

I thought about this for a while, and it's nothing simple. First off, the sun is fuzzy, it is not a point source of light. I suppose we could pretend that the sun was a point source, it still wouldn't be trivial.

[DavidSh]

No, that's not even close to right, half an ellipse would mean that the arcs near the poles were narrowing much more rapidly..

I don't know what you mean here. The interior curve on a crescent moon is the terminator -- the division between the lit and unlit portions of the moon. Assuming the moon is a sphere, the terminator is a great circle. Except during new moon or full moon, this circle is viewed obliquely. A circle viewed obliquely is seen as an ellipse.

Maybe that last part requires explanation. You are familiar with cones? The usual cone you see is a right circular cone, where the base is a circle, and the line from the base to the vertex is perpendicular to the base. If, instead, the line from the base to the vertex is at some other angle, you get a oblique circular cone. The family of curves you get by slicing an oblique circular cone by a plane is the same family of curves you get by slicing a right circular cone by a plane.

Now we can draw the lines from your eye to the terminator of the moon to get an oblique circular cone. The curve you see is the intersection of this cone with the plane perpendicular to the line (the image plane), and so is an ellipse. Or would be, if you could see the entire terminator, instead of just the portion on the face of the moon you can see. Instead you see just half of the ellipse.

The sun being at a finite distance, and being larger than a single point modifies this very slightly. The terminator will still be a circle, just not quite a great circle, so you will see a part of an ellipse tangent to the outer limb of the moon, but it won't quite be half an ellipse. But I expect this to be a very minor effect.

[veti]

I don't know what you mean here. The interior curve on a crescent moon is the terminator -- the division between the lit and unlit portions of the moon. Assuming the moon is a sphere, the terminator is a great circle. Except during new moon or full moon, this circle is viewed obliquely. A circle viewed obliquely is seen as an ellipse.

I agree that there is an ellipse involved, but I don't believe what you see can fairly be described as "half an ellipse". It's less than half, because you can't actually see a full half of the moon.

That's why you don't see a clear meniscus at the edge, because that part of the ellipse is out of view.

Given that you can't see the most distinctive part of the ellipse shape, I think what you can see is as near as makes no real difference to being a small arc of a much larger circle.

[DavidSh]

But how could this ellipse not be tangent to the outer limb of the moon? That seems to be the question, about the "meniscus". By imagining the extreme case where the moon is very close, I can see that the visible section of the ellipse needn't be exactly half.

[shawnhcorey]

You see half of the ellipse (if the Moon was a perfect sphere). Spheres are radial symmetric, so it does not matter the angle you see the terminator at. It will always be half an ellipse, from one semi-major axis to the other.

[DavidSh]

You see half of the ellipse (if the Moon was a perfect sphere). Spheres are radial symmetric, so it does not matter the angle you see the terminator at. It will always be half an ellipse, from one semi-major axis to the other.

That's true if you view from an infinite distance, and is very nearly true from Earth. From much closer it could be a smaller segment of an ellipse, but still tangent to the limb of the Moon (to the horizon, if you are really close to the Moon).

[shawnhcorey]

That's true if you view from an infinite distance, and is very nearly true from Earth. From much closer it could be a smaller segment of an ellipse, but still tangent to the limb of the Moon (to the horizon, if you are really close to the Moon).

I'm not likely to get really close to the Moon tho I would like to.

[halfeye]

Given that you can't see the most distinctive part of the ellipse shape, I think what you can see is as near as makes no real difference to being a small arc of a much larger circle.

I suspect that this is almost it.

The limits are the outer edge of the moon, when the sun is eclipsed or the moon is as new as it can be, and a straight line when the moon is half-full, and back to the edge of the moon when the moon is totally full. The edge of the moon is a circle because the moon is a sphere, and the straight line is an arc of a circle of infinite diameter, the question is what happens in between, are those lines all arcs from increasingly larger circles or something else?

A circle is an ellipse of course, but I don't think that the line is a full half of a non-circular ellipse, because that would imply that the taper from the tips was more exaggerated than what we see, particularly near the half moon there would be very curved horns to the inner edge of the moon, and I don't think I've seen that.

[jayem]

Spoiler: Um Phases not eclipses (left lack of thinking for record)

Show

The thing [earth] creating the shadow is roughly spherical.
The light source is roughly spherical
.

If we:
ignore the moon as a distraction and have a giant static sheet of paper in space
ignore oblate spheroidness
The light source, line to planet, planet, line to shadow, and hence the shadow will be rotationally symmetric around the axis.
Hence the shadow would be a pure circle, the only possible deviations from a platonic circle being that the edge is 'grey', and in fact I think the primary sun size effect is rather that the 'shadow' shrinks at first.

If we have a 2D circular moon, we're now cutting out near the circle (radius moon sized) another circle (with a slightly different centre and radius depending on earth. Still circular.
If we're projecting onto a spherical moon, then the circle will be distorted a little. The shadow at the equator will be just a smidgin bigger (it's nearer the sun and nearer us than the poles, so the effect is doubled). IMO the deviation is negligible.

[ETA except then what about when the moon in >1/2.]

Actually I think I can rescue this. The shadow of the moon on the moon, by similar reasoning must also be circular. WLOG have the sun at (0,0,-Lots), the moon radius 1 at (0,0,0). The moons shadow is thus a circle (sin t, cos t, 0).
(Tangent) If the suns radius is sufficiently big then it is actually a (cross sectional) circle at the point which (viewed from above) has a tangent that reaches the suns edges. Still a circle, but a bit further back (or if the sun was small and near, front).

We're now viewing from an angle, this effectively compresses the x co-ordinates, while leaving the y co-ordinate unchanged. So the inside edge should be a curve of the form (x sin t, cos t), I.E an ellipse. This is literally a conic section (circular 'shadow' projected onto a perspectivey cone ... so makes sense.)

[DavidSh]

Some day I want to take a series of pictures of a lunar eclipse, and stack the pictures to show the full extent of the Earth's shadow. But the last few lunar eclipses for me have either had the Moon behind clouds, or below the horizon.

[halfeye]

Spoiler: Um Phases not eclipses (left lack of thinking for record)

Show

The thing [earth] creating the shadow is roughly spherical.
The light source is roughly spherical
.

If we:
ignore the moon as a distraction and have a giant static sheet of paper in space
ignore oblate spheroidness
The light source, line to planet, planet, line to shadow, and hence the shadow will be rotationally symmetric around the axis.
Hence the shadow would be a pure circle, the only possible deviations from a platonic circle being that the edge is 'grey', and in fact I think the primary sun size effect is rather that the 'shadow' shrinks at first.

If we have a 2D circular moon, we're now cutting out near the circle (radius moon sized) another circle (with a slightly different centre and radius depending on earth. Still circular.
If we're projecting onto a spherical moon, then the circle will be distorted a little. The shadow at the equator will be just a smidgin bigger (it's nearer the sun and nearer us than the poles, so the effect is doubled). IMO the deviation is negligible.

[ETA except then what about when the moon in >1/2.]

Actually I think I can rescue this. The shadow of the moon on the moon, by similar reasoning must also be circular. WLOG have the sun at (0,0,-Lots), the moon radius 1 at (0,0,0). The moons shadow is thus a circle (sin t, cos t, 0).
(Tangent) If the suns radius is sufficiently big then it is actually a (cross sectional) circle at the point which (viewed from above) has a tangent that reaches the suns edges. Still a circle, but a bit further back (or if the sun was small and near, front).

We're now viewing from an angle, this effectively compresses the x co-ordinates, while leaving the y co-ordinate unchanged. So the inside edge should be a curve of the form (x sin t, cos t), I.E an ellipse. This is literally a conic section (circular 'shadow' projected onto a perspectivey cone ... so makes sense.)

Wow, that was some lack of thinking.

I think I've come to the conclusion that in the case of my underthinking it has to be all of a half of an ellipse. I certainly haven't noticed the horns on almost half moons, I suppose maybe I usually see them when they're just over half not just under, or maybe I took it for clouds being in the way or something.

[jayem]

Wow, that was some lack of thinking.

I think I've come to the conclusion that in the case of my underthinking it has to be all of a half of an ellipse. I certainly haven't noticed the horns on almost half moons, I suppose maybe I usually see them when they're just over half not just under, or maybe I took it for clouds being in the way or something.

That was my thinking too. Or if you like a full ellipse (for parallel rays going through the poles) that we only see the near half of (except at new/full moon where both are exactly on the edge).
For non parallel light (big sun or near sun) I think you still see half an ellipse (but especially giving my mix up earlier I'm going to be cautious)

I don't get how people do, "phase/shadows etc... make it trivially obvious it's a sphere". With paper and enough time and thinking you can just about do it. Alternatively with bright lights and reflective surfaces. But even knowing the answers, it's easy to get confused.

[Chronos]

Yes, the horns of an almost-half moon are very narrow, for a short length. The reason you've never noticed them before is that they're very narrow, for a short length. Since the Sun is not a point source, there's some fuzziness to the terminator, and that fuzziness is increased as viewed from Earth by atmospheric effects, so the short, narrow horns get blurred out.

[gomipile]

Yes, the horns of an almost-half moon are very narrow, for a short length. The reason you've never noticed them before is that they're very narrow, for a short length. Since the Sun is not a point source, there's some fuzziness to the terminator, and that fuzziness is increased as viewed from Earth by atmospheric effects, so the short, narrow horns get blurred out.

Also, that small of a feature of the shadow may be of a similar scale as the surface features of the moon.