Will the earth's orbit around the sun eventually decay as a satellites orbit decays around earth orbit?
Satellites orbit around the Earth in circular orbits (usually- a circle is actually just a special case) because they have enough speed that even though they are constantly 'falling' towards Earth due to the pull of its gravitational field. They experience a 'centripetal' acceleration- it is always towards the center of the orbit, which is the center of the Earth (as long as it isn't a HUGE satellite). This is the same model that is used to show how the Earth moves around the sun, even though it's orbit is slightly less circular than most satellites'.
It does not explain, though, why satellites' orbits decay and they fall to Earth. If the only force acting was that of Earth's gravity, and that doesn't change, then every orbit should be the same as the previous one- they shouldn't fall to Earth at all! They do, however, because they do not orbit in a perfect vacuum- the Earth's atmosphere extends very far into space, although it is extremely rarified (not much of it around). This very very small amount of gas acts like any other fluid you know of- water, for example. It acts to slow down the motion of the satellite (because of frictional forces). A slower satellite but the same gravitational force means that the orbit gets closer to Earth- and the density of the atmosphere increases. This denser atmosphere means that the frictional force is greater, so its slows down more quickly, so the orbit decays more quickly. As you might have noticed, this leads into an orbit that decays slowly at first but speeds up until the satellite begins to hit the atmosphere 'proper' and burns up.
That doesn't quite answer the question though- does Earth experience slowing of its orbit because of contact with the Sun's 'atmosphere'. The short answer is no- the sun is so far away and the Earth is so heavy that any force is very very small and any acceleration is far smaller! In fact, if the orbit started to decay, the radiation pressure of all of the light from the sun would increase, which would act to limit the rate of decay.
So overall- probably not.
Michael Inkson, Physics Undergrad-to-be, Cambridge, England
Remembering Newton's Laws of Motion will help understand this, a body will remain at rest or in a state of uniform motion unless acted upon by force. Satellite orbits decay because their mass is relatively small and so little force is required to change their velocity and bring about orbital decay. The Earth has very large mass and is unlikely to encounter sufficient force to slow its motion. The Earth should remain safely in orbit until the Sun becomes a red giant and absorbs it.
'Physicists like to think that all you have to do is say, these are the conditions, now what happens next?'