How is the polar angle of a pulsar beam determined?

Question

I just read the question "What is a typical polar angle of a pulsar beam?" and I was surprised that it had an answer. I thought there was little real information besides the timing of the pulses, and it seems like any given timing could be consistent with a beam with any cone angle.

As I think about it, if you know the angular width of the beam as it leaves the pulsar, then you could infer something from the relative durations of the pulse and the time between pulses. Is that it? Or is it something else?

Note: I misunderstood the first question to be asking about $\alpha$ in the notation of the accepted answer, but they were really asking about $\rho$, the angular radius of the "lighthouse beam". It is easier for me to understand how that could be estimated.

Answer 1

The simple answer is that yes, we can determine the pulse width if we know the angular radius of the emission cone and a couple other geometric quantities about the pulsar and its orientation relative to us.

Pulse profiles are complicated - with some exceptions, they're more than just quick spikes or smooth Gaussian curves. Most involve multiple components added together into complicated features; the beams have structure, and that structure leads to a fingerprint of sorts. Some switch between different emission modes over various cycles. There are plenty of fascinating nuances to study.

Nonetheless, even with these complicating factors, we can indeed define a pulse width. It depends on three things: the angular radius of the emission cone, $\rho$, the angle between the magnetic and rotational axes, $\alpha$, and the impact parameter, $\beta$, which defines the angle between the magnetic axis and the line of sight of the observer.

After a mess of spherical trigonometry, we can arrive at an expression for the pulse width, $W$ (see Gil & Han 1981, or any other work on emission geometry): $$\sin^2\left(\frac{W}{2}\right)=\frac{\sin^2(\rho/2)-\sin^2(\beta/2)}{\sin\alpha\sin(\alpha+\beta)}$$ Therefore, the beam size, the location of the magnetic axis, and the orientation of the observer do uniquely determine the pulse width, as you guessed.

Obviously, it's unlikely that an observer can know these additional parameters entirely a priori, but that doesn't mean we don't know anything about them. You might expect that $\alpha$ would be randomly distributed, and it's possible that pulsars are born as such, but it is expected that $\alpha$ decreases as the pulsar ages due to the same torques that increase the spin period (Young et al. 2010). The two axes align on timescales of $\sim10^{6\mathrm{-}7}$ years. If you could determine the age of the pulsar, and you assumed that it was born with $\alpha\approx90^{\circ}$, you could make some estimates about the likely distributed of $\alpha$ in the present day.