<blockquote>
  <p>Out of 100 pulsars, how many will have a beam that crosses the Earth?</p>
</blockquote>

<p>About twelve. </p>

<p>"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." <a href="https://www.aanda.org/articles/aa/pdf/2004/48/aa0399-03.pdf" rel="noreferrer">M. Kolonko et al.: On the pulse-width statistics in radio pulsars</a></p>

CC BY-SA 4.0
2020-05-05T14:00:17.227Z
2020-05-06T03:19:44.053Z
2020-05-06T03:19:44.053Z
tuomas
<p>The other answers cover the geometric part, but that only tells you what fraction of <strong>pulsars</strong> 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.</p>

CC BY-SA 4.0
2020-05-06T22:36:22.330Z
2020-05-06T22:36:22.330Z
John Doty
<p>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 <span class="math-container">$4\pi$</span> steradians.</p>
<p>The angle swept out on the sky by a pulsar with an emission cone of width <span class="math-container">$\rho$</span> turns out to be</p>
<p><span class="math-container">$$\zeta=4\pi\sin^2\left(\frac{\rho}{2}\right)$$</span>
covering a fraction of the sky
<span class="math-container">$$f=\frac{\zeta}{4\pi}=\sin^2\left(\frac{\rho}{2}\right)$$</span>
The opening angle <span class="math-container">$\rho$</span> can often be deduced from the pulsar's period. Many long-period pulsars obey the power-law model <span class="math-container">$\rho\propto P^{-1/2}$</span>; the proportionality constant is sometimes described piecewise. However, millisecond pulsars tend to deviate downwards from the <span class="math-container">$P^{-1/2}$</span> relation by a factor of a few, as shown in Fig. 12 of <a href="https://ui.adsabs.harvard.edu/abs/1998ApJ...501..270K/abstract" rel="noreferrer">Kramer et al. (1998)</a>:</p>
<p><a href="https://i.sstatic.net/OLpml.png" rel="noreferrer"><img src="https://i.sstatic.net/OLpml.png" alt="Plot of opening-angle period relation" /></a></p>
<p>If you wanted to choose a representative opening angle to determine the probability that a <em>particular</em> 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 <span class="math-container">$\rho$</span>. Picking <span class="math-container">$\rho=40^{\circ}$</span>, for example, gives <span class="math-container">$f\approx0.12$</span>, as tuomas quoted in their answer, which is a reasonable value.</p>

CC BY-SA 4.0
2020-05-07T12:43:17.257Z
Do calculations as these assume no... "wobble"... component of the rotation?
Stian
CC BY-SA 4.0
2020-05-07T15:17:27.933Z
@StianYttervik Correct, I assumed no wobble.
HDE 226868
CC BY-SA 4.0
2020-05-05T17:11:36.780Z
2020-05-05T22:10:54.467Z
2020-06-17T09:47:40.653Z
HDE 226868
<p>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. </p>

<p>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?</p>

<observational-astronomy><pulsar>
What is the probability of seeing a neutron star as a pulsar?
usernumber
# What is the probability of seeing a neutron star as a pulsar?

(Asked by: username_0 on 2020-05-05)

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?

**Tags:** observational-astronomy, pulsar

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## 3 answers

### Answer by username_3 on 2020-05-05. Score: 18 | Accepted answer

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)](https://ui.adsabs.harvard.edu/abs/1998ApJ...501..270K/abstract):

[![Plot of opening-angle period relation](https://i.sstatic.net/OLpml.png)](https://i.sstatic.net/OLpml.png)

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.

> **Comments:**
> **username_4**: Do calculations as these assume no... "wobble"... component of the rotation?  
> **username_3**: @StianYttervik Correct, I assumed no wobble.  

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### Answer by username_1 on 2020-05-05. Score: 18

> 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](https://www.aanda.org/articles/aa/pdf/2004/48/aa0399-03.pdf)


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### Answer by username_2 on 2020-05-06. Score: 3

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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