The north star, Polaris, is neither constant nor particularly bright as many believe. Shakespeare got the poetry right but the astronomy wrong for this star.
"I am constant as the Northern star,
of whose true-fix'd and resting quality
There is no fellow in the firmament.
The skies are painted with unnumber'd sparks,
They are all fire, and every one doth shine;
But there's but one in all doth hold his place:"
With his speech in Shakespeare's play, Julius Caesar indicates he will not change his mind. The northern star is constant; so is Caesar. There is a common misconception that arrogant Caesar compared himself to the brightest star.
Many people think the north star, Polaris, is the brightest star in the night sky. This is incorrect. As a second magnitude star, Polaris is only the 48th brightest. It is however the brightest star in the constellation Ursa Minor, the little dipper.
The brightest star is Sirius, which for mid northern latitudes is to the south. Someone lost at night and looking for the brightest star in the sky to find north is therefore likely to completely reverse the compass directions.
The significance of Polaris is that it is nearly directly above the north pole not that it is bright. Hence it is the north star.
In Shakespeare's reference Caesar will not be moved. So the constancy likely refers to the position of the north star during the night. In this sense it is constant; Polaris does not move over the course of the night.
Polaris is located nearly directly above Earth's north pole. Being closely aligned with the rotational axis, Polaris stays at a nearly constant position. The other stars appear to move in a circular path around Polaris, as Earth spins on its axis.
The north star is however not constant in two ways. Polaris isn't always the north star and Polaris is a variable star.
Watch a spinning top wobble as it spins. The spin axis of the top moves in a small circular path. Earth does the same thing as it spins. Because Earth is much larger and more massive than a top its wobble takes about 26,000 years.
This wobble in Earth's axis causes the north star to change. The point on the sky above the north pole follows a roughly circular path across the sky over 26,000 years. Right now Earth's polar axis points nearly directly to Polaris. It is still moving closer. In 2015 Polaris will be closest to directly above the north pole. Polaris will then begin moving away from the pole.
In a few thousand years Polaris will no longer be the north star, but it will be the north star again in 26,000 years. This cycle is called precession.
Polaris is also a Cepheid variable star. Variable stars change in brightness. Cepheids vary because they pulsate in size. Polaris expands and contracts with a periodic cycle lasting 3.97 days. Cepheids and other pulsating variables increase in brightness as they get larger in size and fade when they shrink.
Polaris is however not a normal Cepheid variable. The amount it changes in size and brightness is decreasing, while its overall brightness is increasing. Astronomer David Turner predicts that Polaris may stop varying in size and brightness. The inconstant northern star is challenging our understanding about how Cepheid variable stars work.
Shakespeare's line: "constant as the northern star" is poetic but astronomically incorrect.
Turner, David et al., The Period Changes of Polaris, Pub. Astron. Soc. Pac., vol. 117, p. 207. 2005.