Just thinking about "galactic habitability zones." Is this even a valid question?
EDIT: I am getting down-voted, and as per the comment below me this seems to be because I failed to answer the question. I thought the question was about Galactic habitability zones as the description mentions but it seems I was wrong. If your question is about the eccentricity of the Sun's orbit around the center of the galaxy, then that question has already been answered here: How can one get the eccentricity of the orbit of the Sun around center of the Milky Way?. I'll leave the rest of my answer in case you were interested in the "habitability" of our galaxy. Cheers.
So "habitable" in the context of our solar system is really talking about radiation flux, where if too much of the sun's radiation is hitting the surface of our Earth, the Earth's atmosphere will heat up too much and the environment would be unsustainable for the development of life. So the Earth is close enough to the sun that the sun has a very strong effect on the dynamics of Earth's environment (obviously, since all of life feeds in some indirect way on the sun!)
However, in terms of the Earth in the context of the Galaxy, I want to clarify that the distance between stars in any galaxy is actually massive compared to the distance between the Earth and the sun. Really, the chance that our solar system will interact with any other star or solar system in the next few billions of years is virtually zero.
With this in mind, I'd say the only consideration about a "Habitable Zone" would be, as someone else has said, our distance from the super-massive black hole at the center of the Milky Way. But since this only makes up a tiny portion of our galaxy, the entire galaxy is really "habitable" for our solar system.
Hope that helps!