The first observed neutron stars were the radio pulsars. They are the only neutron stars we see outside of binary star systems, and in fact they are rarely found in binary systems. Several pulsars sit in supernova remnants, which provides the observational link between neutron stars and supernova explosions.
The radio pulsar is the neutron star as a dynamo. We see radio pulsars for precisely the same reason that we can generate electricity from steam or flowing water with a dynamo: a changing magnetic field generates an electric field that drives a current. In a dynamo, the current is driven through wires passing through the magnetic field. In a pulsar, the current is pulled from the surface of the neutron star and generated through the spontaneous creation of electrons and positrons in the strong electric field. The current in a pulsar generated electromagnetic radiation, particularly radio waves, that can be seen from Earth.
Neutron stars are born spinning rapidly. This is effectively inevitable, because main sequence stars are themselves born spinning. When the core of a massive star collapses, the conservation of angular momentum causes the core to rotate more rapidly, converting gravitational potential energy into rotational energy, and producing a neutron star that can complete a rotation in several tens of millisecond. The maximum rate of spin is set when the centrifugal force equals the gravitational force at the surface of the star. For a 1.4 solar mass neutron star with a radius of 15 km, the maximum rate of spin is around 1000 revolutions per second. In practice, a pulsar rotates at about a revolution per second, with most pulsars completing a revolution in 0.2 to 2 seconds.1 A small handful of very young pulsars spin much faster than this, completing a revolution in several tens of millisecond.
A pulsar's rotational energy is very small compared to the energy released in a supernova explosion. The binding energy of a neutron star, which is the energy that goes into a supernova, is of order 15% of the neutrons star's rest mass. In comparison, a neutron star with a 1 second rotation rate has a rotational energy that is 10-6 of the binding energy of the star. The small energy reservoir fates a pulsar to a short life.
So, how much energy is generated by a pulsar as it spins? Most of the pulsars we see are estimated to be no older than 100 million years based on the rate of change of their rotation period—it appears that older, more slowly rotating pulsars cease producing radio waves. Most observed pulsars are about 10 million years old. This implies that a pulsar with a period of 1 second loses 10-13 of the pulsar's binding energy per year, which is roughly about 10% of the power generated by the Sun.
The simplest theory that connects the power generated by a spinning neutron star to the star's rotation rate imagines the neutron star as a dipole magnet spinning in a vacuum. Such a magnet generates dipole electromagnetic waves—this radiation is called a Poynting flux. One assumes in such a calculation that the magnetic pole is tilted with respect to the rotation pole. Close to the star the magnetic field appears to be nearly a dipole; in this region the magnetic fields themselves appear to be rotating around the spin axis with the star, as though the field lines were wire loops attached physically to the star. As one moves away from the star, this picture breaks down; the field lines begin to bend away from the direction of rotation, as though a wind were blowing our imaginary lines of wire so that they trailed the rotation of the star. This happens at the light cylinder, an abstract cylinder aligned with the rotation axis of the star that synchronously rotates with the star with the velocity of the speed of light. Beyond the light cylinder, the magnetic field lines lag so far behind that they become transverse to a line from the star. These field lines propagate outward at the speed of light, eventually becoming pure electromagnetic waves with a period equal to the rotational period of the pulsar. These waves carry rotational energy away from the pulsar.
This power loss is proportional to Ω4 B2 sin2α, where Ω is the rotation rate of the star, B is the magnetic field strength, and α is the angle between the magnetic field axis and the rotation axis of the star. The first point from this simple calculation is that pulsars lose energy very rapidly when they are born; for this reason, only very young pulsars rotate rapidly. The second point is that the measurement of a pulsar's slowing is a measurement of the pulsar's magnetic field under this theory. The Poynting flux theory gives a magnetic field strength of order 1012 Gauss at the neutron star's surface, which compares to the Earth's magnetic field of 0.3 Gauss. Subsequent observations of x-ray spectra from x-ray pulsars, which are neutrons stars with strong magnetic field in compact binary systems, find similar values for the magnetic field strength of a neutron star.
While the Poynting flux gives a nice, simple estimate of energy loss, its gives no explanation for the electromagnetic waves we see from a pulsar. The radiation we see is not the electromagnetic radiation from a spinning magnetic field. That radiation has a characteristic frequency of 1 Hz, which is many orders of magnitude below the radio frequencies. The radio waves we see are a consequence of the pulsar's strong magnetic field. Repeating what was said before, a changing magnetic field generates an electric field. At the surface of a pulsar, in the absence of charges or currents, the rotation of the magnetic field generates an incredible electric field. The electric field strength is R Ω B/c, where R is the star's radius and c is the speed of light. For a pulsar period of 1 second, radius of 15 km, and magnetic field of 1012G, the electric field is 3 × 108esu. This electric field not only exerts many orders of magnitude more force on a proton than does the star's gravitational field, it can accelerate a proton to 108 times its rest mass energy over a stellar radius, and an electron to 1011 times its rest mass energy.
The generation of an electric field at the star's surface is what drives all of the interesting physics associated with a pulsar. At this point the physics becomes quite complex, because the current generated at the pulsar surface modifies the electric field driving the current and the magnetic field generating the electric field. Our initial assumption that the region above a pulsar's surface is a vacuum is wrong, because the electric field that gets generated in the vacuum must drive a current that alters the electric field. The precise structure of the magnetic and electric fields and of the electric current is an outstanding problem in astrophysics. The important point, however, is that an electric field of sufficient strength to drive a current is generated by the pulsar's rotating magnetic field.
One final bit of exotic physics: if an electric field becomes too large, it spontaneously generates electrons and positrons that extract energy from the electric field. In other words, the electric field decays into matter and antimatter. This process appears to provide much of the current in a pulsar, rather than the extraction of electrons and protons from the star's surface.
The current generated by a pulsar flows away from the star along the magnetic field lines at the magnetic poles of the star. Because the field lines are curved, the charges in the current follow a curved path as they move away from the star. This motion causes the charges to radiate electromagnetic waves, called curvature radiation, at radio frequencies. Because the emission in in the direction of the field lines, which are pointing away from the star's poles, the radio waves are emitted in a beam aligned with the magnetic axis. As the star rotates, the orientation of the magnetic axis changes relative to Earth, and the radiation from the pulsar appears to pulse.
Most observed pulsars are several kpc away, although some pulsars are as close as 100 pc from Earth. Clearly, nearby for a neutron star is not nearby at all. This reflects both the rarity of high-mass main-sequence stars relative to low-mass main-sequence stars and the short lifetime of a radio pulsar.
The best-known pulsar is the Crab pulsar (PSR 0531−21), which sits near the center of the Crab nebula. The supernova that created the Crab pulsar was observed in 1054 AD, so the Crab pulsar is just short of the millennium of its birth. Because of its youth, the Crab pulsar carries more rotational energy that most other radio pulsars. It rotates once every 0.033 seconds, which is an order of magnitude shorter than most other pulsars. This high rotation rate provided some of the initial evidence that pulsars were too small to be white dwarfs; a white dwarf with this rotation rate would fly apart, and a white dwarf pulsating at this rate would violate the speed of light limit. The striking supernova remnant surrounding the pulsar is powered by the pulsar. Without the energetic particles injected into the nebula by the pulsar, the Crab nebula would have faded away long ago. The light from the remnant is therefore directly tied to the slowing of the Crab pulsar's spin. The pulsar is estimated to be 2 kpc way.