How fast can neutron stars spin?

Image Credit: NASA (artist impression)

The fastest spinning stars in the universe are called millisecond pulsars because they rotate between 100 times and 1000 times a second!

These neutron stars are the best natural clocks in space and if we want to understand the universe it's only sensible to want to know what limits their rotation rate and what could be the most precise clock of this type.

The first millisecond pulsar was discovered in 1982, the year I was born in. I know.... I am getting old, but so is this discovery. This pulsar spins about 641 times every second and was able hold onto its speed record until 2006 when scientists discovered a faster pulsar that has a spin frequency of 716 Hz. By 2006 I was 24 and already getting close to finishing my Cornell PhD on the nonlinear development of the R-mode instability, a mechanism that provides a limiting spin frequency for neutron stars, which depends on their internal physics, and could explain why we see so few fast pulsars and why we have not seen any that spin much faster than 700 times a second. 

I work on other topics now, but the topic of my PhD thesis has remained close to my heart.  So, below I summarise my latest paper in collaboration with Prof. Ira Wasserman (one of my two PhD advisors) on R-modes. The paper is now published in the Astrophysical Journal and on the arXiv at http://arxiv.org/abs/1305.2335 .

Brief Summary

The conventional picture is that a sharp boundary layer forms causing high viscous dissipation. In this case, the R-mode instability prevents the spin-up of neutron stars to frequencies larger than about 300 Hz, where gravitational driving matches viscous dissipation due to boundary layer viscosity. The red line in the figure below shows the typical evolution of a neutron star in the frequency (Hz) - temperature (in units of 10⁸ K) plane in the presence of boundary layer viscosity. Since we observe neutron stars than spin faster than 300 times a second, this picture cannot be correct or complete. 

We argue that boundary layer viscosity becomes negligible when  the transition between the fluid core to the crystalline crust happens over distances larger than 1 cm.  We find that the spin up from accretion torque is larger than the spin down due to emission of gravitational waves and that continued spin-up due to accretion from a companion allows frequencies of up to about 750 Hz. The blue line represents a neutron star trajectory in the frequency- temperature plan in the latter case.The maximum frequency would be lower (plausibly about 750 Hz) than in the figure for typical Copper pair cooling inside the neutron star core.

What are R-modes? What is the R-mode instability? How does it work?

When a sharp boundary layer forms.

The star continues to spin-up (low viscosity).

R-modes are stellar oscillations that exist in rotating neutron stars and are driven unstable by gravitational radiation emission. Fast spinning neutron stars remain R-mode unstable as long as the gravitational driving is larger than the viscous dissipation.

In the two figures, the viscous dissipation and gravitational driving are equal on black curve; above the said black curve the L=m=2 R-mode is unstable and below the curve the R-mode is stable and its amplitude damps exponentially to insignificant values.

 In the instability window, where the L=m=2 R-mode is unstable, gravitational radiation emission spins down the star. However, the star can continue to spin up if the spin-up torque caused by accretion from a companion is greater that the spin-down caused by gravitational radiation and by magnetic dipole radiation emission. This happens when when viscosity is very low. In this scenario, the L=m=2 oscillation mode saturates at low amplitude due to nonlinear coupling to near-resonant oscillations in the star. The R-mode saturation amplitude is proportional to the viscous dissipation of the pair of near-resonant modes that are first excited by the R-mode. The limiting spin frequency of the star depends on accretion physics, viscosity and neutrino cooling. In this paper we include the first analytic estimate of this frequency and of the saturation amplitude of the R-mode that includes nonlinear effects.

Potential Future work
1. It would be good to understand the temperature distribution of the neutron stars that could have active R-modes.  R-modes heat up the star. But is that observable? Do they heat it too much to be consistent with existent data or are the error bars so large that we cannot say much?

2. The distribution of couplings between the R-mode and other modes in the star is not known, and the approximations currently made are very rough. The couplings can be computed analytically for certain very specific models. They assume that the neutron is incompressible, which mean its density is assumed to be constant everywhere, and find very sparse couplings between modes. It would be interesting to see how the coupling distribution changes when the  incompressiblity constraint is relaxed. My guess is that the coupling distribution is more complex, and may be either sparse or dense depending on parameters.

Would R-modes affect the stability of the "neutron star clock"? how about glitches?
The R-modes I have considered are oscillations of the core of the star. Active R-modes heat the core of the star. However, connecting the surface temperature to the core temperature is not trivial. Oscillations in the frequency of the star due to R-modes are un-modelled on short timescales.

R-modes on Earth?

R-modes (known as Rossby waves) also exist in the oceans and the atmosphere of Earth and can cause changes in the weather pattern. However, such oscillations in the core of the star are proportional to the angular frequency of the star and are only significant for neutron stars rotating close to millisecond periods.

What are neutron stars and how to we observe them?

Neutron stars are the tiniest and densest stars in the universe. When a star that is about ten times as massive as our sun reaches the end of its life, it explodes violently ejecting its outer layers, but leaving the core. The left-over core then collapses under its own weight to a neutron star or a black hole. The neutron star will be about once or twice as massive as our sun, but packed into the diameter of an average city (about 10 to 20 km) and will spin very fast. Its density and temperature is far greater than anything achievable on Earth. Since these stars are so different from our own planet, our understanding of the physics of this ultra-dense matter is still poor.

Telescopes on Earth were the first to see pulsars, which are a rotating neutron stars that emit radio signals. This signal is spotted if it is beamed towards us. It appears to pulse because the star rotates. Jocelyn Bell Burnell identified the first pulsar in 1967. They were first thought to be signals from aliens in space and hence dubbed Little Green Men. A few years later they were identified as rapidly rotating neutron stars. Now we see neutron stars with many telescopes and observatories on Earth and in Space. Scientists time them accurately and try to infer their surface temperature when possible.

Observational windows opening in the future

Scientists use gravitational wave detectors on Earth  (e.g., LIGO, VIRGO) to search for gravitational waves emitted by single neutron stars either due unstable oscillations inside them or due to small mountains on their surface (continuous wave searches) and by binary neutron stars (compact binary searches). The binary neutron stars are particularly important gravitational wave sources because we know they exist and we know where some of them are.  We cannot yet say the same for the binary black hole sources, or the black hole-neutron star binaries.

Advanced LIGO will be taking data and hopefully discovering some of these sources in about two-three years from now. So far there are only null results to gravitational wave searches that put upper limits on existing physics. This is the LIGO outreach page that describes some of the most important publications written by the collaboration. 

Pulsar clocks

Isolated millisecond pulsars are really good clocks.  Potential applications of accurate pulsar timing include using pulsars to guide space craft navigation and gravitational wave detection in a different frequency band than ground based detectors.

I will conclude with a pulsar version of "twinkle-twinkle little star":

"Twinkle, twinkle little star
You must be a small pulsar
Far away from Earth you spin
You have high magnetic field
Twinkle, twinkle little star
Degenerate matter's what you are."

Note: This is a slightly modified version from SMBC comics.

Literature.  
  1.  R. Bondarescu, and I. Wasserman, “Nonlinear Development of the R-mode Instability and the Maximum Rotation Rate of Neutron Stars”, Astrophys. Journal 778, 9 (2013). [arXiv:1305.2335]. 

2. R. Bondarescu, S. A. Teukolsky, I. Wasserman, “Spinning Down Newborn Neutron Stars: Nonlinear Development of the R-mode Instability”, Phys. Rev. D 79, 104003 (2009).  [arXiv:0809.3448]. 

3. R. Bondarescu, S. A. Teukolsky, I. Wasserman,  “Spin Evolution of Accreting Neutron Stars: Nonlinear Development of the R-mode Instability”,  Phys. Rev. D 76, 064019(2007) [arXiv:0704.0799].