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It's often said that the pulses of the pulsar have a very precise period, which helps to detect easily exoplanets that cause fluctuations in the time of arrival (TOA) of the pulses. Anybody knows why?

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Pulsars are precise because... in essence, there isn't much to cause them to be imprecise. It's as if you spun a top in intergalactic space and left it there for a hundred years - it would likely still be spinning (at pretty much the same rate).

In essence, pulsars are super-dense neutron stars that are rotating really fast, and their emission axis happens to line up correctly with Earth so that we see it as a pulsating source of radiation. Because there isn't really much to slow their rotation down, it stays relatively constant (though there can be what we call pulsar glitches). However, pulsars do gradually "spin-down", and their period decreases over time (very slowly, mind you), and we can calculate this - we call it the p-dot, or period-derivative - the rate of change of the period, if you will. (As James K pointed out in his comment, this emission of radiation is also part of what causes this spin-down.)

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    $\begingroup$ The "why" is that the angular momentum can only be carried away by EM radiation from the spinning magnetic dipole, which, as you say, is slow. $\endgroup$ – Keith McClary Jan 12 at 1:49

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