Waseda Institute for Advanced Study (WIAS)Waseda University

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Using gravitational waves to elucidate the physics of rotating neutron starsL
OKAWA Hirotada, Assistant Professor

My interest in gravitational-wave astronomy

The mechanisms of many phenomena in the universe are unknown. To study phenomena like that, we create a model based on theory and then calculate the behavior of the model. This is called simulation: if the phenomena are reproduced by the simulation, we consider that the model is correct.

I have been doing a simulation to explore the mechanism of the “supernova explosion” that occurs when a massive star reaches its end. A supernova explosion happens when a shock wave generated near the center of a massive star reaches the outer edge of the star. Adopting the influential “neutrino heating mechanism” which sustains the shock wave to propagate outwards, I have attempted to reproduce that kind of explosion by performing simulations using Japan’s “K computer.”

I am currently interested in gravitational-wave astronomy. Gravitational waves are phenomena that occur when any body moves. The distortion of space-time that is, the change of gravity propagates as light speed waves that ripple outward from the source. We can use gravitational waves to directly observe astronomical phenomena that cannot be observed using electromagnetic waves (e.g., black holes merging, or the center of intense astronomical phenomena). This is expected to be another important key to exploration of the newly born universe.

Gravitational wave detection and theoretical calculation

The existence of gravitational waves was predicted by Einstein in his general theory of relativity in 1916, but the first direct observation was done very recently, in September 2015. Gravitational waves resulting from binary black hole mergers were observed by LIGO (Laser Interferometer Gravitational-wave Observatory) in the United States. Gravitational waves have been observed several times since then, and in 2017, gravitational waves generated during the merger of neutron star binaries were observed for the first time.

Observation data from scenarios including gravitational waves also contains a huge amount of noise (irrelevant data), so it is necessary to extract actual gravitational wave signals using many theoretical waveforms to detect those waves (Fig. 1). Once we can detect gravitational waves, we will be able to obtain detailed information about the stars before they coalescence. However, current observational accuracy is not capable of providing much information about the period between the merging of stars and the time when a spinning star settles down—so one of the motivations of my research is to theoretically calculate the rotational motion after coalescence and obtain useful information by analyzing the observed data.

Figure 1. Data from observation of gravitational waves resulting from binary black hole mergers observed in 2016 Source: Gravitational Wave Open Science Center, (https://www.gw-openscience.org) The horizontal axis is time, the vertical axis is amplitude (which represents space-time distortion), and the wave interval is frequency. The red line represents the observed data and the blue line represents the theoretical waveform. Gravitational wave detection requires observation at two locations; the data shown here was detected at Hanford. The waveform corresponds to the motion of the black holes shown at the top of the graph. In the case of black holes merging, it is relatively easy to calculate the theoretical waveform that will appear after the merger, but in the case of neutron stars merging, it is difficult to calculate the resulting waveform because that depends strongly on unknown information such as the equation of state for the matter.

Proto-neutron star generated by a supernova explosion

Another interesting phenomenon occurs after a supernova explosion, when a proto-neutron star, the origin of a neutron star, is born. However, in order to become a neutron star, the proto-neutron star must emit neutrinos and gravitational waves and be cooled for tens of seconds. It is not realistic to model such long-term phenomena in detail with a dynamic simulation; rather, a method based on stepwise calculation would be efficient. In other words, we need to repeat the steps, first to calculate the number of neutrinos that escape from a star with a certain equilibrium shape, then to construct a star in an equilibrium shape after taking into account the number of neutrinos that have escaped, and finally to calculate the number of neutrinos that escape from that star.

However, there is still no method for constructing rotating stars in equilibrium under general conditions.

Development of W4, a new calculation method

One common aspect of the above two problems is the analysis of rotating neutron stars based on general relativity. To do that analysis, however, we must solve non-linear elliptic partial differential equations. Those equations can be used to yield non-linear simultaneous equations by means of the finite difference method or the spectral method. Until now, the Newton-Raphson method (NR method) has been used for numerically calculating non-linear simultaneous equations. However, the NR method has the disadvantage that it cannot solve a system if the initial value is far from the solution. It has another negative aspect in that an initial value close to the solution must be prepared in order to find the solution in the first place.

Working on that problem, I came up with a calculation method called W4. The NR method is used when current information and the previous information (differential) yield the next information. The W4 method adds the information that precedes the previous information (two differential). When I tried to solve a relatively simple equation using the NR method and the W4 method, starting from various initial values, there was a wide range of initial values that could not be reached with the NR method, but with the W4 method, even if I started the calculation from the initial values of such a region, I was able to obtain a solution.

Fig. 2. Comparison of NR and W4 methods. The solution of the non-linear simultaneous equation shown in the upper left is the intersection of the red circle and the two blue curves in the figure at lower left. The upper right presents the solution using the NR method and the W4 method. The two figures on the lower right show the results when the calculation is started from an initial value in this range, using both the NR method and the W4 method. In each case, x is a solution, and the color coding of red, blue, orange and green indicates which of the four solutions is obtained. Using the NR method, the solution cannot be reached from the initial value of the region shown in black; however, using the W4 method, we can obtain any of the four solutions regardless of the initial value.

I think that by applying this calculation method to the calculation of general relativity, it may be possible to determine the equilibrium shape of rotating stars. My current calculations are only preliminary ones, but the results are quite satisfactory. In the future, I would like to observe gravitational waves theoretically from bodies such as rotating stars, and derive the relevant physical information.

Interview and composition: Keiko Aimono
In cooperation with: Waseda University Graduate School of Political Science J-School

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