About the Image
The Alpha Centauri system is not visible from much of the northern hemisphere. The below image shows this star system and other objects near it in the sky.
Image Credit for Alpha Centauri photo: Copyright Akira Fujii / David Malin Images.
How Do We Calculate Distances of This Magnitude?
The methods astronomers use to measure distances to the stars are pieces of fundamental and active work in astronomy with important implications for how we understand the Universe around us.
One of the most accurate methods astronomers use to measure distances to stars is called parallax. If you hold your finger in front of your face and close one eye and look with the other, then switch eyes, you'll see your finger seem to "shift " with respect to more distant objects behind it. This is because your eyes are separated from each other by a few inches - so each eye sees the finger in front of you from a slightly different angle. The amount your finger seems to shift is called its "parallax".
Astronomers can measure parallax by measuring the position of a nearby star very carefully with respect to more distant stars behind it, then measuring those positions again six months later when the Earth is on the opposite side of its orbit. If the star is close enough to us, a measurable parallax will be seen: the position of the star relative to the more distant background stars will have shifted. The shift is tiny - less than an arcsecond even for the nearest star. (An arcsecond is 1/60 of an arcminute, which is 1/60 of a degree.) (Imagine the Universe has more information on calculating parallax.)
Image Credit: Imagine the Universe, NASA/GSFC
Why Are These Distances Important To Astronomers?
Stars are not actually stationary objects! The Galaxy is rotating, and the stars are in orbit around its center. Not every star moves at the same rate - how fast they orbit can depend on where the star is located within the Galaxy. Our Sun, being fairly far from the Galactic Center, takes over 200 million years to circle the Galaxy once. Some of the stars near us are moving faster than us, and some slower. As Phil Plaitt, from Bad Astronomy says, "...like cars on a highway, stars continually pass each other as they orbit the Galaxy. They change positions, slowly, but measurably."
Image Credit: Frog Rock Observatory, public domain and copyright-free.
Why Can't We Travel Faster Than the Speed Of Light?
There's no fundamental reason why we can't get as close to the speed of light as we like, provided we have enough energy. But this is probably far in the future.