Is antumbra part of shadow darker than penumbra part of shadow?

I'm exploring different types of shadows casted by objects.
I want to know if antumbra part of a shadow is darker than penumbra part.

I've found two misleading pictures on Wikipedia:
First: Second:

Which one is correct?

This example outside astronomy shows that antumbra is lighter than penumbra part of the shadow:

btw. Are there any reliable online references on that topic?

Answer: At a given distance from the sun, the antumbra is always darker than the penumbra. Explanation: Within the antumbra, the shadow is uniformly dark. The depth of the antumbra shadow reflects only the ratio of the angular size of the two bodies. Where the antumbra meets the penumbra, there is more and more sunlight as more and more of the sun becomes visible. At the outer edge of the penumbra, the sun is completely visible, and there is no longer any shadowing. This is a smooth transition and there is not a huge instantaneous drop in brightness at the "edge" of the penumbral shadow. (The entire penumbra is really the "edge" of the shadow, transitioning from no shadow to maximum shadowing within the antumbra.)

In one sense there is no such thing as either a antumbra or penumbra. Neglecting minor diffraction effects, a point source of light is either visible from a given point or it is not visible from that point.

In order to deal with an extended light source, like the Sun, we need to answer two questions to determine the light level at a point.

First, what fraction of the Sun's surface is visible from the location in question? We are all in the antumbra of Venus or Mercury during a transit of the planet, but few of us would ever notice. The vast majority of the Sun's surface is unblocked during such a transit, and the light level here is only minutely affected.

Secondly, we need to look at the brightness of those visible parts of the light source. A person who sees, say 1% of the Sun's surface at one point leading up to a total eclipse (penumbra) is seeing points close to the limb, and some points not so close. Someone viewing 1% of the Sun's surface during an annular eclipse (antumbra), is seeing points that average closer to the limb. Since points closer to the limb are dimmer (see the image of the Sun in your first illustration), the second person would experience a dimmer light from the 1% value than expected.

I think that last picture shows shadows cast by two light sources (the reflection in the window glass, perhaps?) Because the length of the umbra is just too short and the angles wrong for it to be due to a small, distant light source such as the sun. Perhaps an overcast day.

Furthermore, I think there are parts of the penumbra that are darker than the antumbra, and vice versa. You could probably qualify the question by specifying that it is a fixed radial distance from the light source.

I'm going do do the annoying teacher thing where I answer a question with a question. Which is darker, a partial solar eclipse, in the penumbra of the moon, or a planetary transit, in the antumbra of an interior planet?

TL;DR the first example is quite accurate. Yes, the antumbra is darker.

This question also bugs me for like over two decades when I read astronomy introduction books at school library... Why antumbra is darker than penumbra? IIRC the illustration back then comes with fairy rough details, something like the lower picture. It looks so unnaturally to the point of unbelievable.

Plus my simple experiment with a torch gives contradiction. So I just think the book is misprinted and move on to other interesting topics... Until recently something brings me back to this curiosity again.

But I abandon astronomy discipline for so long, all I know for living now is writing programs, so I comes up with this simple simulator, and here is a result:

The sun is to the left outside of this picture (at around $$x = -100$$). The moon is at $$x = 0$$ with diameter of $$y = \{20..30\}$$, and cast total eclipse shadow in umbra region from $$x = \{0..25\}$$. Note that we may observe two gray rays as tails from the end of intense shadow that seems to be darker than surrounding area. There is no such rays, they are illusions that our eyes try to trick our brain!

The result is similar to example no. 1 and I'll try to elaborate why this result should be correct. First, examples and experiments outside of astronomy is irreverent since light scattering on earth, while in space the main light source is only from the sun.

So we starts with a model of the sun that is very far away from us so it visible size stay the same no matter we measure it inside our interesting region. And for simplicity we let visible size of the moon stay the same when measure at any $$y$$ on the same $$x$$ distant, but vary when change $$x$$.

Let's consider umbra region, within this region the moon is visible larger than the sun and block all of the light. So it is total eclipse, or $$density = 0\%$$.

Now on antumbra region, the moon is visible smaller than the sun, so it can not block all of the light. If we stay at the same $$x$$ and move along $$y$$ within antumbra region, the moon will have constant size, and can block same amount of light from the sun. So light density is constant for any observation point on the same $$x$$.

Lastly the most complex penumbra region, pick a point $$p$$ on the edge of antumbra region and jot down light density of that point (for example, $$density = 50\%$$). next consider a point in umbra region near an edge of the moon ($$x = 0$$), by shifting a little bit outward from the moon (let $$x$$ stay the same), we will start to see some light from the sun. We stop shifting at point $$q$$ when we get $$q$$ to have same density as point $$p$$. Draw a straight line between $$p$$ and $$q$$, this line represent same density of shadow in penumbra region.

So the light/shadow density of umbra-penumbra-antumbra can be roughly view with this contour:

We can see that at a same density, a corresponding contour (dashed line) looks like a cup; starting on one side of the moon, goes straight down until met antumbra's edge, then goes parallel to the moon to get out of antumbra, and lastly go straight up to another side of the moon.

This is just a simple model, in the real world size of the moon and sun will not always stay the same. Density line in penumbra is not a perfectly straight line. Light ray lost its power along the way they travel. And since the sun is a sphere (not a circle), maybe some part of them is brighter than other? All these details should have been take into account if we want more precise model. But for now I think this model is quite good enough to explain the concept.

Recap:

• Only umbra region have the same light/shadow density.
• The other two regions does not say much about density, it main purpose is telling type of eclipse we'll get.
• Antumbra region have same density on the same layer, where each layer is parallel to the moon.
• Penumbra region is the most complex place to calculate density, take it together with antumbra and it will look like a cup.

Think of yourself on the dark side of the earth away from the sun. As you move further away from the earth you eventually get a ring of sunlight around the earth. You are then entering the antumbra. As you get further away from the earth the ring of sunlight gets larger and you see more of the sun. At the same time you are moving away from the sun so the amount of light you are receiving from the sun is smaller so the light gets dimmer. Eventually you see mostly the suns light with very little blocked by the earth.

• This doesn't seem to answer the question, which is about whether the antumbra is darker than the penumbra. Feb 19, 2014 at 1:29

In the penumbral area, part of the disc of the moon obscures a section of the sun. In the antumbral area, the entire disc of the moon obscures a, logically, larger section of the sun. Therefore the antumbra cannot be brighter than the penumbra during the same event.

• Counterexample: which is darker, a partial solar eclipse, in the penumbra, or a planetary transit, in the antumbra?
– rob
Oct 3, 2014 at 18:57
• Rob - that's why I said "during the same event" Oct 4, 2014 at 23:56
• Your answer with a question makes the inquiry meaningless. Of course some penumbra can be brighter/darker than some separate antumbra! Oct 5, 2014 at 0:02