How are millisecond pulsars formed, and what is the evolutionary track of a millisecond pulsar?
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The standard model for the formation of millisecond pulsars is the recycling scenario.
The starting point of this model is a stellar binary system consisting of a neutron star and a secondary 'regular' star, where the neutron star is a normal pulsar with a rotational period of roughly $\sim7$ seconds.
Because a neutron star is such a compact object, its gravitational influence is very large ($F_g \propto M/R^2$). If the two stars are close enough, the outer layers of gas may be pulled of the companion star and flow towards the neutron star. All this matter will settle into an disk structure around the neutron star and start orbiting faster and faster as it draws closer to the surface ($v_{orbit} = \sqrt{GM/R}$).
By the time this matter touches down on the neutron star surface its orbital period can be as fast as a millisecond, certainly significantly faster than the neutron star. This bit of motion (or more precisely, the angular momentum) is then transferred to the neutron star (this process is called accretion), causing it to rotate slightly faster (spin-up).
After some $0.1\sim0.2 M_{\odot}$ (where $M_{\odot}= 2 \times 10^{30}$ kg gives the mass of the Sun) of material is transferred to the neutron star, its rotational period will have decreased from a few second down to milliseconds.
The transfer of this much matter takes a very long time, roughly $10^8$ to $10^9$ years. After this time, and with all the redistribution of mass, the binary system is expected to widen which causes the mass transfer to stop. As the region around the neutron star clear up, the radio emission mechanism turns back on (this mechanism does not work during the accretion stage) and we are left with a millisecond pulsar.
From this point on the neutron star will behave like a regular pulsar: over the course of a very long time it will convert its rotational energy in radio emission and slow down until it becomes to slow to act as a pulsar.
As for the evidence for this model; such binary systems containing an accreting neutron star emit massive amounts of X-ray emission (hence called X-ray binaries) and have been known to exist since the 60's, long before the first millisecond radio pulsar was discovered (in 1982). This entire scenario was proposed quickly after, but could not be confirmed for a very long time.
It was only in the late 90's that a neutron star was found whose X-ray emission was pulsed at millisecond periods (see this, and this), confirming that these neutron stars are indeed spinning up. The final piece of evidence; a millisecond pulsar which switches back and forth between X-ray and radio emission was only found very recently.
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$\begingroup$ Perhaps add the $\dot P-P$ diagram, to be able to visually track the evolution of a MS-pulsar. $\endgroup$ Aug 9, 2014 at 15:24
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$\begingroup$ Really good answer, @MichaelB. Question, though: Wouldn't the system lose energy via gravitational radiation, thus bringing the two stars closer together? Or is that compensated for by the effects of the mass transfer? $\endgroup$– HDE 226868 ♦Aug 11, 2014 at 15:25
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$\begingroup$ @HDE226868 - You're absolutely right, gravitational wave emission could lead to an in spiral, but this is only relevant for extremely tight binaries. For the situation considered here the evolution is completely dominated by mass transfer. $\endgroup$ Aug 12, 2014 at 13:14
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$\begingroup$ Okay, @MichaelB, thanks for clearing that up. $\endgroup$– HDE 226868 ♦Aug 12, 2014 at 16:27