Would it be possible for the atmosphere of a planet to reach beyond its planetary ring so that earth-like biology could develop on some of the asteroids there?
This question is mainly about whether there could be a life sustaining atmosphere there, not if developing life there would be possible (even though in a larger scale this is obviously what I'm interested in knowing).
The problem with this idea is that air resistance would quickly slow the orbit of the asteroids, and bring them crashing down to the surface, it might work if the asteroids were out at geosynchronous orbit but then the atmosphere would just escape.
There are two ways to get a similar effect to what you are looking for though, both of which have already been discussed in fiction:
Rocheworlds
Two tidally locked planets just outside the Roche Limit can orbit each other and share a combined atmosphere. You would be able to fly from one to the other without ever leaving the atmosphere and objects placed at the lagrange points would be able to remain there.
This concept has been explored in the books Flight of the Dragonfly/Rocheworld by Robert Forward and Land and Overland by Bob Shaw.
(http://diasparys.deviantart.com/art/Prometheus-Over-Roche-World-369286493)
Smoke Rings
The Smoke Ring is the atmosphere of a gas giant pulled into a cloud around a neutron star. There are a number of issues with the stability and likeliness of the system but the concept for the most part does hold up as scientifically viable. This idea is explored in some books by Larry Niven, starting with The Integral Trees.
Yes, although it would take a pretty extreme case and it's not going to be long-term stable:
As others have said, the asteroids must be in synchronous orbit or drag would bring them down. Others have dismissed this due to atmospheric bleed or atmospheric pressure but I don't believe these objections are insurmountable.
1) Pressure. Our world is spinning very fast, the equator is pretty close to synchronous orbit already. Note that this world is nothing like round and has a very low surface gravity. As gravity is much lower the pressure gradient is likewise much shallower--you can have a lot more atmosphere without crushing the surface.
2) Bleed. This is a much bigger problem to overcome but it's not insurmountable. We need a much more massive planet than the Earth, one where the difference between synchronous orbit and an escape orbit is on a par with the 11 km/sec between our atmosphere and an escape orbit.
I can't imagine how such a world could occur other than the work of some extremely powerful engineers but there's nothing impossible about it.
Note that the storms on the world itself will be incredibly violent. We have a huge differential in atmospheric speed between the equator (at near synchronous orbit) and the poles (stationary.)
Short answer: No.
If asteroids were within a life sustaining atmosphere, then they would experience enormous drag. Unless they were high enough to be in geostationary orbit, but then the atmosphere would escape very quickly.
It seems tempting to think that a very dense atmosphere could provide buoyancy, but this poses two problems:
Just as an addendum to what everyone else has said, No, this isn't possible, for quite a few reasons.
Lets do some maths though, to explain why.
Lets assume this planet is like earth, but with a ring system, the ring material orbits at the height of the atmosphere, which we'll call 70km above sea level, which combines with earth's radius for an orbital height of 6470km.
In a stable orbit,
$$v^2=\frac{GM}{r}$$
That is to say Velocity squared = Mass of Gravitational Body(Earth) multiplied by the Gravitational constant, divided by the height of orbit. (People interested in deriving this should google hyperphysics orbital mechanics)
Therefore, $$v^2= \frac{(6.67 \times 10^{-11}) \times (5.97 \times 10 ^{24})}{(6.47 \times 10^6)} = 6.15 \times 10^7$$ So $v=7842m/s$. That's about 28000km an hour, or 16900mph
Now, lets consider the fact that average planetary ring is made up of tiny chunks of ice. The Space Shuttle's average descent speed during reentry is about 17500mph, and you see how hot reentry gets. The ice would melt instantly. Even if it was rock (which it almost certainly wouldn't be) the atmospheric friction would create temperatures that would sterilize the rock.
I hope this helps.
I'll take a completely different approach from what everyone else is saying, because everyone (okay, sans ivy_lynx) seems to be assuming that there are asteroids in planetary rings. This is not true. The objects that make up a planet's rings can come in a range of sizes, but none are as big as an asteroid. There are shepherd moons in the rings of some of the solar system's gas giants, but they are very small moons.
The objects within a gas giant's rings are way to small to hold onto an atmosphere. Fortunately, your question doesn't make that assumption. While these small bodies could not have a gravitationally bound atmosphere, the planet they are orbiting could, and so it is this atmosphere we can assume they are traveling through.
So you would have a bunch of small objects rushing through a cloud of gas (and probably dust from the rings). It wouldn't be too conducive to life. The rings are pretty stable, but there are always chances of collisions. There's a big difference between a moon in a stable orbit and a piece of a ring orbiting inside a ring.
I can go into more detail if you want, but I'm in a bit of a hurry, so all I'll say now is that the area would be too chaotic, and the bodies would be too small to have a chance of being a safe harbor for life. But more information is on the way!
I don't think that an atmosphere can extend that far because if gravity is holding it to a planet and said gravity becomes strong enough, it would instead compress the gases into a liquid instead.
But! You could possibly have a supermassive gas giant or brown dwarf and place the asteroids inside its atmosphere. I remember reading that this is possible, but I don't have a link at hand. Their sky wouldn't be blue of course and they might experience a very different day-night cycle if at all, but it is within the extreme theoretical limits.
Alternatively, you could just place dwarf planets in there.
Your main problem is that atmosphere is held where it is through gravity. The strength of gravity affects the gases in the atmosphere. For example Earths gravity is strong enough to hold onto Oxygen and Nitrogen but not Helium and Hydrogen. For an Asteroid to be within Earth's atmosphere enough to react with its gases, it would also be under the effect of Earth's gravity which would lead to, well, bad things.
What you could do is say the asteroid is so dense that the gravitational force it provides is enough to have its own micro-atmosphere, so not TECHNICALLY sharing our atmosphere, but capable of holding one of its own.
Through this, it could obtain gases that have escaped your ring world and built them up over time to have its own.
It could not happen for a standard planetary ring that is in orbit, due to drag. But it isn't impossible to form a structure that appears to be a ring from entities floating in the atmosphere, not in orbit. For example, a titanic number of zeppelins flying about en-masse over the equator at various altitudes, using their motors to avoid being scattered. If the gasbags were reflective, it might look impressive from a distance.
Why people would want such a thing is another question.
