# What is the probability of seeing a neutron star as a pulsar?

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):

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.

• Do calculations as these assume no... "wobble"... component of the rotation? May 7, 2020 at 12:43
• @StianYttervik Correct, I assumed no wobble. May 7, 2020 at 15:17

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