Re: How large would a celestial object need to be to alter the earth's orbit?

I'm doing research for a short science fiction story I'm writing, I'm wondering if there were an object possibly the mass of a large asteroid which would orbit the earth much like the 99942 Apophis then how much influence would it exert over the course of say a century?

An object of any size that actually orbits the Earth doesn't affect its orbit (except to the extent that it is actually the barycenter of the two objects that orbits the Sun). The Moon is a good example of this.

Asteroids don't typically orbit the Earth (and 99942 Apophis does not and will not). The Earth's orbit is affected by any object that passes it, but in order for the effect to be significant, the passing body would need to be both massive and very close. Apophis is a small asteroid, no more than a few hundred meters across, and wouldn't have a significant effect on our orbit even in the extreme case of an impact.

Even the largest asteroid, Ceres (which these days might be considered a dwarf planet), only has 1/10,000th the mass of the Earth, so a close encounter would probably not have a large effect on Earth's orbit. In order for the effect to be significant, you'd probably need to consider planet-to-planet interactions, where the invading body was at least a few percent of the mass of Earth.

Three-body calculations are difficult (in the case of the Earth, you really have to consider this a four-body problem: the Sun, Earth, Moon, and asteroid). A large asteroid passing close to the Earth would result in some sort of momentum transfer- mostly orbital momentum, and possibly some rotational momentum as well. Angular momentum is conserved, but that doesn't mean each body would have the same amount after the encounter as before. Unless the asteroid is huge, it will leave the Earth with minimally more or less orbital momentum, meaning that the Earth would end up very slightly closer or farther from the Sun- not likely enough to affect the climate. An interaction with a body of similar mass, however, could radically change the Earth's orbit- enough to make most life impossible. An even more massive planet could even add enough angular momentum to the Earth to give it escape velocity, and fling it out of the Solar System.

I'd suggest you Google "orbit simulator" and check out the many online and downloadable programs available. This will let you experiment with different mass bodies interacting with the Earth.