Pulsars are neutron stars that emit a beam of electromagnetic radiation that is not aligned with its rotation axis. If the Earth passes through that beam of radiation, we see a pulsar. Pulsars are only observable if the beam crosses the observer's line of sight. Otherwise, we can only see a regular neutron star.

For a pulsar with random orientations, what is the probability of seeing its beam from Earth? Out of 100 pulsars, how many will have a beam that crosses the Earth?


The probability of seeing pulsed emission from a neutron star is simply the fraction of the sky covered by the beam, i.e. the beam solid angle divided by $4\pi$ steradians.

The angle swept out on the sky by a pulsar with an emission cone of width $\rho$ turns out to be

$$\zeta=4\pi\sin^2\left(\frac{\rho}{2}\right)$$ covering a fraction of the sky $$f=\frac{\zeta}{4\pi}=\sin^2\left(\frac{\rho}{2}\right)$$ The opening angle $\rho$ can often be deduced from the pulsar's period. Many long-period pulsars obey the power-law model $\rho\propto P^{-1/2}$; the proportionality constant is sometimes described piecewise. However, millisecond pulsars tend to deviate downwards from the $P^{-1/2}$ relation by a factor of a few, as shown in Fig. 12 of Kramer et al. (1998):

Plot of opening-angle period relation

If you wanted to choose a representative opening angle to determine the probability that a particular pulsar will sweep its beam across Earth, it might be best to pick an angle calculated from the pulsar's period. On the other hand, if you care about a population of pulsars with a random period distribution, you would be better off simply looking up a mean value of $\rho$. Picking $\rho=40^{\circ}$, for example, gives $f\approx0.12$, as tuomas quoted in their answer, which is a reasonable value.

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    $\begingroup$ Do calculations as these assume no... "wobble"... component of the rotation? $\endgroup$ – Stian Yttervik May 7 '20 at 12:43
  • $\begingroup$ @StianYttervik Correct, I assumed no wobble. $\endgroup$ – HDE 226868 May 7 '20 at 15:17

Out of 100 pulsars, how many will have a beam that crosses the Earth?

About twelve.

"The beaming fraction f , that is the mean value of the fraction of observable pulsars or the mean probability of observing a normal pulsar, is 0.124 ± 0.004." M. Kolonko et al.: On the pulse-width statistics in radio pulsars


The other answers cover the geometric part, but that only tells you what fraction of pulsars are seen as such from Earth. The other issue is what fraction of neutron stars are pulsars at all. If they don't pulse, and they don't do something else conspicuous like accrete from a binary companion, neutron stars are very difficult to find. A common rough estimate is that only 1% of neutron stars in our galactic neighborhood are detectable, but this is very uncertain.


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