# How can a pulsar slow down?

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 is it that gravitational waves are emitted and carry away the angular momentum and slow down the rotation, just like the case of orbital motions slowing down binary neutron star or black hole systems?

• "I think they still cannot carry away net angular momentum but just net energy." -- Why do you think this? Commented Apr 27, 2022 at 16:26
• Because the two beams of radiations are in opposite directions. So the angular momenta of the photons should cancel out and the net angular momentum carried by the radiations should be zero. Commented Apr 27, 2022 at 16:30
• Is it that because these are synchrotron radiations and so the two beams have angular momenta in the same direction? Commented Apr 27, 2022 at 16:34
• @velutluna, the synchrotron radiation is produced by the electrons moving in the pulsar's strong magnetic field. They do contribute to the total radiation from the pulsar, but don't have a significant effect on the pulsar itself.
– Cham
Commented Apr 27, 2022 at 16:53
• @Cham So angular momentum loss by gravitational waves is the main reason leading to the slowing down of pulsars? In that case my textbook is wrong then, although the two beams do carry net angular momentum away. Commented Apr 27, 2022 at 16:57

## 2 Answers

Electromagnetic 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 (the usual model for a neutron star magnetic field) 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.

• Gravitational waves are very likely to be a factor, if the pulsar has some bulges (i.e. ellipticity, or deviation from perfect sphericity) and rotates around another axis. For example, the strong magnetic field could distord the spherical shape, giving the pulsar some bulges at the magnetic poles.
– Cham
Commented Apr 27, 2022 at 16:48
• @Cham The distortions are extremely minute, and continuous wave searches by LIGO have put firm constraints on any GW emission. Not sure what the latest constraints are from O3 (!), but even early observations put limits of GW contributions to spin-down of <1% for several notable pulsars. Commented Apr 27, 2022 at 16:58
• Really nice answer. Do you think you can answer my question here: physics.stackexchange.com/questions/705850/… Commented Apr 28, 2022 at 3:17

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.

• Can I say that it's because the radiations are synchrotron radiations and the directions of the magnetic fields at the two poles are the same, and so the angular momenta of the radiations do not cancel out. I guess it's true that thermal radiations of normal stars can only carry away net energy but not net angular momentum. Commented Apr 27, 2022 at 16:40
• @velut luna, the two angular momenta radiated from the two magnetized poles aren't canceling, they're adding up.
– Cham
Commented Apr 27, 2022 at 16:43
• @velutluna In the models, the synchrotron radiation doesn't come from the poles, but from the vicinity of the "speed of light cylinder", where the field from the poles would be rotating at the speed of light if it maintained dipole geometry. Commented Apr 27, 2022 at 17:43
• How does the angular momentum from solar wind work? Isn't the wind mechanism in contrast with the answer in this question? Commented Sep 24, 2023 at 12:48