Alan C. Calder


Neutron Star Mergers

My work on the binary neutron star coalescence problem is with Doug Swesty and Ed Wang. This complex problem brings together several branches of physics and astronomy, including gravitational wave astronomy and nuclear astrophysics. This work is part of the NASA HPCC Grand Challenge of merging neutron stars.

General relativity predicts that binary systems of compact objects will emit energy in the form of gravitational radiation, and that this loss of energy eventually will lead to the coalescence of the system. This process was confirmed by the observation of PSR 1913+16, a binary neutron star system of which one member is a pulsar. The observed rate of decrease of the orbital period of this system is in good agreement with predictions made by general relativity.

Plot of density, increasing as the color moves from yellow to blue to red, showing the merger of two neutron stars. The neutron stars are modeled as initially spherical $\Gamma = 2$ polytropes. Credit for the image, as well as warmest thanks, to Dave Bock at NCSA. Click here for details about the image and here for details about the visualization group project.

Coalescing binary systems are expected to release energy on the order of their gravitational binding energy, $\approx 10^{53}$ erg, and the gravitational waves produced in these events are expected to be observed by gravitational wave detectors currently under construction. The limited range of frequencies of such detectors makes in-spiraling compact binaries the only sources of gravitational waves expected to be observed with high accuracy, and theoretical templates of the expected signal are required to extract signal information from the noisy background. Post-Newtonian methods are adequate for the prediction of waveforms for the early stage of the in-spiral, but predictions in the later stages of the merger, when tidal effects and neutron star structure become important, requires a full three-dimensional numerical simulation.

NSMs are also of interest because the gravitational waveforms produced by these events may yield information about both the equation of state and maximum mass of neutron stars. Additionally, the dynamical behavior of the post-merger waveform may indicate the final state of the coalesced object, neutron star or black hole, and place limits on the initial total mass of the system. The vast amount of energy NSMs are thought to release may be larger than estimated gamma-ray burst energies, $\approx 10^{51}$ to $10^{53}$ erg, thereby suggesting NSMs as a source of the observed gamma-ray bursts should the bursts prove to have a cosmological origin. Simulations of NSMs can test the consistency of the energetics and time scales with the estimated energies and observed time scales of the bursts.

Images from a Newtonian simulation at different time steps showing the inspiral and coalescence of a pair of initially spherical neutron stars:

There are two principal approaches to investigating NSMs. These are models that consider Newtonian gravity with weak field (quadrupolar) or post-Newtonian corrections for the effects of gravitational radiation, and fully relativistic hydrodynamics models with dynamically calculated spacetimes. Our research group is addressing both approaches, and my research has focused on Newtonian and post-Newtonian models. Significant progress can be made with Newtonian and post-Newtonian simulations, and comparing Newtonian-like simulations with fully-relativistic simulations will determine how significant general relativistic effects are in the gravitational waveform of the merger. From this comparison we hope to gain a better understanding of the limits of Newtonian and post-Newtonian theories.

Our initial work focused on obtaining stable Newtonian orbits of polytropic stars for a careful study of the Newtonian tidal instablilty first reported by Rasio and Shapiro (1992). Our study led to the development of V3D, a multidimensional hydrodynamics code for self-gravitating astrophysical systems based on the ZEUS algorithm (Stone and Norman 1992). We found that for $\gamma = 2$ polytropes a tidal instability exists only for very slightly separated binaries, and that we were able to obtain stable orbits (some simulations ran for tens of orbits) for binaries initially separated above the threshold of the tidal instability. We found that numerical effects can mimic the effect of the tidal instability, but that these can be understood and eliminated. These effects included the choice of reference frame of the simulations and the time ordering of the advection and source updates, and we also found that equilibrium initial conditions were essential to maintain stable orbits for narrowly separated binaries. One striking feature seen in our simulations and illustrated in the figure is the formation of tidal arms during the coalescence which transport a substantial amount (several tenths of a solar mass) of material into a rapidly rotating disk surrounding the coalesced object. The image consists of a color-mapped slice plane showing the logarithm of the density in the orbital plane and a ray-traced volumetric rendering of lower density material surrounding the central object.

Movies! Click here to see a movie of a Newtonian coalescence. Click here to see a movie of a Newtonian coalescence beginning from an equilibrum contact-binary configuration. The details of the image above apply to these movies. The equilibrium configuration went for more orbits than the non-equilibrium configuration even though the stars were closer together initially. This result demonstrates the sensitivity of the simulations on the initial conditions.

Currently we are studying post-Newtonian coalescing models. In this case we include a gravitational radiation reaction source term at the 2.5 post-Newtonian order as prescribed by Blanchet, Damour, and Sch\"{a}fer (1990). We are finding that this radiation reaction has a very significant effect on the evolution of the binary system. Comparison between a purely Newtonian merger (the merger occuring because the stars were initially in contact and hence below the threshold for the tidal instability) and a post-Newtonian merger from the same initial conditions shows that the post-Newtonian coalescence occurs in a significantly shorter time, introducing a phase difference in the gravitational waveforms. We also find that the final coalesced objects are different in structure and have a different total angular momentum. Thus we are finding that the inclusion of a post-Newtonian radiation reaction plays a very important role on the dynamics of the merger, and this result further motivates the need for fully relativistic simulations. Recently with our visualization expert, Dave Bock, we have been studying the structure of the gravitational wave radiation produced during the coalescence. We are finding that the gravitational wave radiation is produced mostly in the tidally deformed regions of the stars.

The images show the density color map through the coalescing stars and an isosurface of the gravitational wave radiation at $10^{35}$ erg/gram/cm^3 at several times during the course of a simulation. Click here for more details about the images and for movies of the simulations.

Another study in progress is a comparison of purely Newtonian simulations performed with different hydrodynamics methods. The results of our efforts in obtaining stable Newtonian orbits, that purely numerical effects can change the results of a particular simulation, and the disparate conclusions of several research groups about the presence of dynamical instabilities indicate the importance of careful comparison of simulations performed with different numerical techniques. To this end, we are comparing the results of simulations with our code to the results of simulations from the same initial conditions performed with Piecewise Parabolic Method hydrodynamics and Smooth Particle hydrodynamics. We are particularly interested in monitoring the ability of each method to conserve angular momentum, and the results of this study will have implications for simulations of white dwarf mergers and binary star systems as well as NSMs.

F. A. Rasio and S. L. Shapiro Ap. J., 401 226 (1992)

J. M. Stone and M. L. Norman 1992, Ap. J. S. 80 791 (1992)

L. Blanchet, T. Damour, and G. Schafer, M.N.R.A.S., 242 289 (1990)

Click here to see the UIUC News Tip about our work. Astronomy magazine also ran the news tip on their web page with some graphics. Click here to see the article with images.

Click here to see the article on our work that appeared in NCSA's Access magazine. Click here to see a more recent article about some of the new visualizations we have created with the help of Dave Bock.

Click here for a recent manuscript on our NSNS research. The manuscript was published in the proceedings of the Second ORNL Symposium on Nuclear and Atomic Astrophysics.


Astrophysical Thermonuclear Flashes
Neutron Star Mergers
Core Collapse Supernovae


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Last modified: December 12, 2006