In the early 80's, Taylor and Hulse (Nobel prize winners for this work) discovered a system (PSR1913+16) where one pulsar circles another compact object. Because the pulsar pulses occur at very regular intervals, they can be used as a clock. Moreover there are several physical effects which can be used to determine the shape of the orbits of the pulsar and the compact object. It was found that these objects are slowly spiraling into each other, indicating that the system is losing energy in some way.
This system (more about PSR1913+160) can also be studied using the General Theory of Relativity which predicts that the system should radiate gravitational waves carrying energy with them and producing the observed changes. These predictions are in perfect agreement with the observations. This is the first test of General Theory of Relativity using objects outside our solar system.
A pulsar is an extremely dense star, so dense that all of the empty space in an atom has been squezzed out and one is left with the individual atomic nuclei touching each other. Under these circumstances, the star is of the order of 1015 times denser than ordinary matter. In this state most of the protons in the nuclei have combined with atomic electrons to form neutrons and the star is therefore also known as a neutron star.
The binary pulsar PSR 1913+16 was discovered by Hulse and Taylor in 1974. This is a pair of neutron stars orbiting each other, with a semi-major axis of 2.3 light-secs. By measuring the orbital motion, one can deduce much information about the pulsars themselves.
An extraordinary feature of this binary system is that the extreme density and quite small orbital radius means that the orbital precession is an astounding 4.2 degrees per year! The measured value agrees very well with Einstein's theory.
Indeed, this binary system has proven to be a rich testing ground for gravitational theories. One interesting aspect is that the period of the orbit is gradually speeding up as the pulsars spiral in towards each other. This implies a loss of energy -- compare it to an artificial satellite gradually spiralling in to the Earth due to atmospheric drag -- and the rate at which this is happening is consistent with the radiation of energy via gravitational waves -- a vibration of the curvature of spacetime -- in complete accord with the predictions of Einstein's theory.