Re: Would a ring around Earth cast a shadow on the surface?

Good question! It turns out that do not really need to really speculate so much as to simply look at Saturn because the effects you ask about are the same for giant planets and terrestrial planets alike. Well, provided we accept a few assumptions.

Assumptions

We will need to make a few assumptions. In effect, these are a way of stating that the rings around our imaginary terrestrial planet are like Saturn's main rings: The A, B, and C rings. (The ones you see through your backyard telescope.)

1. The rings orbit in the planet's equatorial plane. We think that it would be difficult to get rings that do not do so, so this is a reasonable assumption.
2. The rings orbit between 1.25 and 2 planetary radii from the planet's center. This is about where all of the ring systems around the giant planets exist. The outer limit exists because if you try to form a dense ring much farther from the planet and the ring particles want to clump into moons too much. This sweeps up the ring material and destroys the ring.
3. The ring system is flat. It is difficult to directly measure the thickness of the Saturnian main rings for a few reasons, but we think that the rings are thin, a few tens of meters thick at most, perhaps only a few meters thick in some areas. Let us assume that the same is true of our terrestrial ring.

Question One: Would the Rings Be Visible?

So, let's answer your two questions now. First, could we see the ring from the surface of the planet? The answer is, "Yes, from most of the planet." You can see the outer edge of the rings from the surface of the planet from anywhere closer to the equator than 60 degrees north or south latitude. (It just takes a little high school trigonometry to show this for yourself. Just remember that your local horizon is at a right angle to the local radius line to planet's center and you will be fine.) To see the entire main ring system, you would need to be closer to the equator than 36 degrees north or south latitude. (Do the same math, just use the inner edge of the rings now.) Before you think that most of the planet cannot see much of the rings, remember that most of the area of a sphere is near its equator not its poles (they are bigger around there... think about the area of a strip that is a meter wide just to the south of the equator and a strip a meter wide just to the south of the North Pole), not to mention the fact that most of us like to live nearer to the equator. So overall, I would predict that most people could see at least part of the rings, if not all of them.

There is one tiny little catch here, however: people at or very near the equator will not be able to see the rings. At least, they will not see them very well. The trouble, you see, is that the rings are so very, very thin. Seen edge-on, they essentially disappear. Saturn's rings appear to do this as seen from Earth twice in each of its 26-year orbits. The Cassini spacecraft has gotten this view far more often since it passes through Saturn's equatorial plane in each of the spacecraft's orbits of Saturn. How far out of the equatorial plane do you have to be to see the rings well? The answer appears to be a few degrees, based on what I've seen from Cassini images. There is no one magic number where you suddenly get a good view, though, so how far away you would want to live is a matter of taste.

By the way, the rings would also be visible at night. The parts of the rings which were not in the planet's shadow would be brightly lit against the night sky and very pretty. The parts which were in the planet's shadow would be visible where they blocked out the background stars. In the daytime, the rings would still be visible, although the daylight would wash them out somewhat, rather like it washes out the Moon.

Question Two: Would the Rings Cast a Shadow?

And now, the second question: Would the rings cast a shadow on the planet? To answer this, we need to make one more assumption. This assumption is not about the rings, but about the planet: I will assume that the planet has a tilt like Earth's or Mars's. (Coincidentally, this is similar to Saturn's tilt. I know of no reason why this could be anything other than pure chance, however.) For tilts like those, for most of the year the rings would cast large shadows on the winter hemisphere of the planet just like at Saturn. And you thought it got cold now! (OK, some light would filter through, depending on how thick the rings were. But it may not be much for a dense ring like the B ring.) Near the equinoxes, the rings would cast almost no shadow at all, only a thin little sliver right near the equator as the rings would be nearly edge-on to the Sun. Saturn is moving to that configuration now and will get there in 2009, by the way.

If the planet has too little tilt, the rings will always be near this equinox configuration and there will never be much shadow to speak of. If, on the other hand, the planet has a very large tilt, the ring's shadow could actually miss the planet completely for whole seasons of the year: consider the case of Uranus which spins nearly on its side. For part of the year, it keeps one pole facing the Sun all day long. The shadow of the rings never comes close to the planet in this configuration. However, for the purposes of science fiction this situation is probably unlikely because such planets would make bad places to try to live, given a reasonable alternative.

About those Assumptions

What about those assumptions? Well, if the rings were not actually in the equatorial plane of the planet, it would not matter to my answer as it turns out. It would just chance how we refer to the latitudes from where the rings were visible. (It would depend on the latitude, longitude, and time. And that is why I assumed that the rings were in the equatorial plane.) If the rings were closer to the planet, they would be more difficult to see from most of the surface (the planet gets in the way), if they extended farther from the planet than I assumed, they would be even more visible. Finally, if the rings were a lot thicker than I assumed, they would not disappear from the equator and they would cast better shadows around the equinox.

Did Earth Have a Ring?

Since you brought it up, I thought that I would mention that our current best theory for the formation of the Moon suggests that the Earth was struck a glancing blow by a large planetesimal. This, we think, flung a lot of material into orbit around the Earth. Our Moon could have accreted from this debris disk and then migrated away (the disk would have been located close to the Earth, within a few Earth radii). Does that debris disk sound like something familiar? Yep, it would have been a sort of a ring. (In fact, some of the principle researchers in the field started out researching Saturn's rings and got into the Moon's formation more or less by accident, or so they tell me.) So the Earth did probably once have a ring. Sadly, since we had just been struck by a huge impactor which probably left most of our surface molten and since the planet was probably a few tens of millions of years old, there really was no one around to enjoy the view.

Resources

Most of this answer has been based on direct analogy to Saturn. Since Saturn has almost the same tilt as Earth, we really only need to look at the images to get a pretty good idea of what the shadows would look like on an Earth-like planet. You can find quite a few views of Saturn with the shadow of its rings at the Cassini Imaging Central Laboratory for OPerationS (CICLOPS), including the image I've shown here and its caption which describes the context of the image more. (Disclosure: I work for CICLOPS, so I am sort of plugging our site. But I do not work on the public site, so I am not really bragging for myself, honest.)