I know this has probably been debated to death, but it's really buggging me:

What's the speed of gravity???

Isn't it the same as the speed of light?

There is no speed of gravity. The acceleration is just under 10metres/sec^2 , which applies until the object reaches termial velocty on the earth.

If the Sun hypothetically ceased to exist in an instant, will the Earth continue an orbit for 8 mins before heading off on a tanget?

No, and the time that light takes to get here from the sun certanly has nothing to do with it

So what would happen to the path of the Earth in the above scenario?

Nick:

The Earth would continue revolving around what was once the sun until the lack of gravity reaches it.

How long does that take? Would you see the light of the sun "go out" beforehand?

The influence of the suns gravity would beging decreasing immediately, but the influence wouldn't instantaneously stop altogether.

The bit I've never understood is:

If gravity propagates at the speed of light, how does gravity escape from a black hole if light doesn't?

Nickelodeon is correct. Gravity propagates at the speed of light. However, relativistic effects (gravitomagetism and frame-dragging) make it appear to travel faster. The Newtonian approximation (in which gravity propagates instantaneously) is very close to correct.

From Does Gravity Travel at the Speed of Light?

In general relativity, on the other hand, gravity propagates at the speed of light; that is, the motion of a massive object creates a distortion in the curvature of spacetime that moves outward at light speed. This might seem to contradict the Solar System observations described above, but remember that general relativity is conceptually very different from Newtonian gravity, so a direct comparison is not so simple. Strictly speaking, gravity is not a "force" in general relativity, and a description in terms of speed and direction can be tricky. For weak fields, though, one can describe the theory in a sort of Newtonian language. In that case, one finds that the "force" in GR is not quite central--it does not point directly towards the source of the gravitational field--and that it depends on velocity as well as position. The net result is that the effect of propagation delay is almost exactly cancelled, and general relativity very nearly reproduces the Newtonian result.

So what is the path? Does its' orbit "shallow" out gradually until it travels off in a tangent?

Yes pretty much, until the momentum has ran out, eventually it come to a stop relative to where the sun once was.

In general relativity, gravity is a characteristic of spacetime; it is not a force. Gravity doesn't need to "escape" a black hole the way a force-carrying particle does.

Both questions (does gravity travel at the speed of the light and how does gravity escape from a black hole) are asked quite often. Much better answers than mine can be found in these FAQ at sciastro.astronomy.net.

Comes to a stop??? This is the 2000 year old Aristotelean view of physics.

Dont be a dick. I was refering to the frame of reference of the spot it once orbited around i.e. the sun. It's not going to continue on a tangent when the tendency of the suns gravity has worn off so therefor without a force acting upon it it will become relatively stationary . Of course it is always moving relative to some frames of reference, but the sun is in that same frame of reference.

OK, so if the the frame of reference is the backround stars?

Nickoldeon:

I am fairly certain that the majority - if not the total extent - of the gravity would take as long a time as the sun poofing in the sky. That is, once the sun disappears, Earth goes flying.

In that frame the Earth will continue moving. What force makes it come to a stop? Momentum is a conserved quantity. It does not "run out".

this is a bit complicated
gravity influences time so it is a bit hard to say if a gravity field somehow disappears how fast this lack of field spread.

practically this is a lot more complicated, the sun can not disappear, at best it can move away at near light speed. the energy required for it to suddenly accelerate to near light speed would itself cause an immensly greater gravity field than the suns field etc.

dunno...