Sizing Up Neutron Stars with Gravitational Waves

Gravitational waves from colliding neutron stars have improved our understanding of the interiors of these fantastically compressed objects and helped us measure their radii. How much more precisely will we be able to measure neutron stars with future gravitational wave observatories?

Measuring a Neutron Star

Hubble Space Telescope image of a neutron star

This image from the Hubble Space Telescope shows a neutron star. It was estimated to be no more than 28 kilometers (16.8 miles) across and have a temperature of 1,200,000℉ (670,000℃). [Fred Walter (State University of New York at Stony Brook) and NASA/ESA; CC BY 4.0]

When stars more massive than about eight times the mass of the Sun explode as supernovae, they often leave behind a neutron star: the rapidly spinning, magnetized remnant of the star’s core. Neutron stars are immensely dense and strong, packing more than the mass of the Sun into a sphere the size of a city. Counterintuitively, the more massive the neutron star, the smaller it is. Exactly how a neutron star’s size varies with its mass is described by its equation of state: the relationship between mass, radius, and density.

Already, observations of gravitational waves from colliding neutron stars have helped us hone our estimates of the neutron star equation of state. A 1.4-solar-mass neutron star — around the lower limit of a neutron star’s mass — will have a radius between 10.5 and 13 kilometers. Researchers suspect that future gravitational wave observations will narrow this range further, and new work explores how precisely we’ll be able to measure neutron stars in the future.

Computing Collisions

To probe this question, Daniel Finstad (University of Washington; University of California, Berkeley; Lawrence Berkeley National Laboratory) and collaborators Laurel White and Duncan Brown (both Syracuse University) simulated the gravitational waves produced by many pairs of colliding neutron stars.

plot of neutron star mass versus radius

Soft, medium, and stiff equations of state (blue, orange, and green lines, respectively), as well as the full set of equations of state used in the analysis. [Finstad et al. 2023]

The team modeled three populations of colliding neutron stars with different equations of state, labeled “soft,” “medium,” and “stiff.” These equations of state cover the range of neutron star interiors currently allowed by observations. The stiffness of the equation of state affects both the neutron stars’ sizes and how much they’re deformed by tidal forces as they approach a collision. These changes leave an imprint on the gravitational waves produced in a collision, allowing us to extract the equation of state from gravitational wave observations. Modeling a range of equations of state is important because our ability to measure a neutron star’s equation of state depends on the equation of state itself; “soft” interiors produce signals that are fainter than “stiff” interiors do.

Upcoming Observations

With a suite of simulations in hand, Finstad’s team modeled what future gravitational wave observatories would detect if faced with these synthetic signals. They considered future upgrades to the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo detectors that would bring them up to their maximum sensitivity as well as the proposed Cosmic Explorer, which would have arms 10 times as long as LIGO’s and therefore be more sensitive.

plot showing how long it would take LIGO–Virgo to measure the neutron star equation of state to within 2%.

Number of years needed for LIGO–Virgo to observe enough mergers to measure the neutron star equation of state to a precision of 2%. Results are shown for stiff, medium, and soft equations of state (green, orange, and blue, respectively), as well as for different values for the neutron star merger rate, shown with the timescales at the top. [Finstad et al. 2023]

Finstad and collaborators found that an upgraded LIGO-Virgo would be able to measure the neutron star equation of state to within a precision of 1.9–0.7%, depending on the stiffness — but it would take 10, 20, or 57 years to observe enough mergers of stiff, medium, or soft neutron stars (respectively) to reach that precision. Cosmic Explorer, on the other hand, would require only a year to amass a similarly large collection of observations, measuring the equation of state to within a precision of 0.56% or better.


“Prospects for a Precise Equation of State Measurement from Advanced LIGO and Cosmic Explorer,” Daniel Finstad et al 2023 ApJ 955 45. doi:10.3847/1538-4357/acf12f