How can a pulsar slow down?

Question

I saw in some astronomy textbooks that pulsars gradually slow down due to the loss of energy by its radiation. I wonder why this is possible? Although the radiation is now not thermal but in the form of two beams, I think they still cannot carry away net angular momentum but just net energy. So while energy decreases the angular momentum should be conserved. The only way to have decreasing energy but constant angular momentum is to have increasing momentum of inertia by $$E=\frac{L^2}{2I}$$ which is clearly not what happens to a pulsar. So should it be the gravitational waves emitted that carries away the angular momentum and slow down the rotation, just like the case of orbital motions slowing down in binary neutron star or blackhole systems?

Answer 1

Radiation can indeed take away angular momentum. Thinking about this just from a classical point of view the flux of energy carried by electromagnetic waves (in vacuum) is $$ \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}\ ,$$ where $\vec{S}$ is the Poynting vector. The linear momentum associated with this is given by $$ \vec{p} = \epsilon_0 \int \vec{E} \times \vec{B}\ dV\ = \frac{1}{c^2} \int \vec{S}\ dV\ , $$ where the integral is over a volume and the angular momentum is $$ \vec{L} = \frac{1}{c^2} \int \vec{r} \times \vec{S}\ dV\ .$$

If the waves are not perfectly transverse - and the total fields of say a rotating magnetic dipole are not - then angular momentum can be carried away.

Gravitational waves are unlikely to be a factor at all since gravitational wave radiation only arises from any time-dependent quadrupole mass moment and this will be zero if the neutron star has axial symmetry. aLIGO does now have observational limits on this for some pulsars - for example the rate at which the Crab and Vela pulsars are losing rotational kinetic energy now can only have contributions of $<0.2$% and $<1$% resepectively from gravitational wave radiation (Abbott et al. 2017).

It is of interest to look back at Ostriker & Gunn (1969) who give a detailed examination of the rotation-powered pulsar model, where they do consider that both a rotating magnetic dipole or a rotating mass quadrupole could be responsible for the spin-down of neutron stars (via electromagnetic and gravitational wave radiation respectively). Although it appears that the Crab and Vela pulsars are mainly braked by electromagnetic radiation now, it is worth noting that whilst the rate of change of angular momentum due to electromagnetic radiation scales as $\omega^3$, the equivalent losses for gravitational waves scale as $\omega^5$. That means that any gravitational wave spin-down could have been much more important at higher (by factors of $\geq 10$) spin rates than observed for the Crab and Vela pulsars now.

Answer 2

If the pulsar slows down, its angular momentum decreases. This implies that there's some angular momentum radiated away. Rotational energy decreases too, of course. There could be several mechanisms that radiate away the angular momentum and rotational energy of a pulsar. Most notably:

1. Dipolar electromagnetic radiation from the polar beams.
2. Quadrupolar gravitational radiation, if the pulsar has a non-spherical shape.
3. Mass loss from solar wind.

The rotational kinetic energy is $$\tag{1} K = \frac{1}{2} \, I \, \omega^2. $$ The angular momentum is $L = I \, \omega$. Both $I$ (moment of inertia) and $\omega$ (angular velocity) can vary, depending of the mechanism at play. Gravitational potential energy must also be taken into account, if the mass $M$ and the radius $R$ are variable: $$\tag{2} U = -\, \frac{kGM^2}{R}, $$ where $k$ is a dimensionless constant that depends on the internal structure of the pulsar.